3.1 w-standard sets

Let us fix a set w and a standard set I such that $w \in I$ .Footnote ⁵

In particular, if $\nu \in \mathbb {N}$ is nonstandard, then $w = \nu $ is good and, more generally, $w = \langle \nu , z \rangle $ is good for any set z.

The following facts are immediate consequences of known results (see [Reference Kanovei and Reeken14, Sections 3.3, 6.1, and 6.2, esp. Theorem 6.2.6]). For easy reference we give the proofs in Section 5. Quantifiers with the superscript $\operatorname {\mathrm {st}}_w$ range over w-standard sets.

An equivalent formulation of (1) is Transfer into w-standard sets:

$$ \begin{align*}\exists x\; \phi(x) \rightarrow \exists^{\operatorname{\mathrm{st}}_w} x\; \phi(x).\end{align*} $$

Another equivalent formulation for $\phi (v_1,\ldots ,v_r)$ with w-standard parameters is

$$ \begin{align*} \forall^{\operatorname{\mathrm{st}}_w} x_1,\ldots, x_r\; [\,\phi (x_1,\ldots, x_r)\longleftrightarrow \phi^{\operatorname{\mathrm{st}}_w} (x_1,\ldots, x_r)], \end{align*} $$

where $\phi ^{\operatorname {\mathrm {st}}_w}$ is the formula obtained from $\phi $ by relativizing all quantifiers to $\operatorname {\mathrm {st}}_w$ . In yet other words, $(\mathbf {V}, \mathbf {S}_w, \in )$ satisfies Transfer. It also satisfies Boundedness and, for good w, Bounded Idealization (see [Reference Kanovei and Reeken14, Theorem 6.1.16(i)]), but not Standardization (ibid, Theorem 6.1.15). Although we do not need these results in this paper, together they show that $(\mathbf {V}, \mathbf {S}_w, \in )$ satisfies all the axioms of $\mathrm {BST}$ except Standardization. Note that here $\mathbf {S}_w$ plays the role of a new “thick standard universe.” On the other hand, $(\mathbf {S}_w , \mathbf {S} , \in )$ satisfies Transfer, Boundedness, Standardization, and Countable Idealization; here $\mathbf {S}_w$ plays the role of a new “thin internal universe.”

Idealization can be strengthened from $\mathbb {N}$ to sets of cardinality $\kappa $ if w is chosen carefully. We leave the technical definition of $\kappa ^+$ -good sets to Section 5.2 (see Definition 5.7). For our applications we need only to know that for every standard uncountable cardinal $\kappa $ and every z there exist $\kappa ^+$ -good w so that z is w-standard, a result which is also proved there.

Proposition 3.5 (Idealization into w-standard sets over sets of cardinality  $\le \kappa $ ).

Let $\phi $ be an $\in $ -formula with w-standard parameters. If w is $\kappa ^+$ -good, then for every standard set A of cardinality $\le \kappa $ ,

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}\! \operatorname{\mathrm{fin}}} a \subseteq A \;\exists y\; \forall x \in a\; \phi(x,y) \longleftrightarrow \exists^{\operatorname{\mathrm{st}}_w} y \; \forall^{\operatorname{\mathrm{st}}} x \in A \; \phi(x,y).\end{align*} $$

Propositions 3.4 and 3.5 do not exhaust the properties of the universes $\mathbf {S}_w$ ; for a list of further useful principles see [Reference Kanovei and Reeken14, Theorem 3.3.7].

3.2 Coding external sets

Definition 3.1 provides a natural way to represent w-standard sets by standard functions: A w-standard $\xi \in \mathbf {S}_w$ is represented by any standard $f \in \mathbf {V}^I$ such that $f(w) = \xi $ . Note that every $\xi $ has a proper class of representations. This causes some technical difficulties (see Section 5), which for our purposes are best resolved by fixing a universal standard set V so that all objects of interest are subsets of V or relations on V; usually one requires $\mathbb {R}\subseteq V$ . By Representability, for every $\xi \in V \cap \mathbf {S}_w$ there exists a standard $f \in V^I$ such that $f(w) = \xi $ . Moreover, if $\psi $ is an $\in $ -formula with standard parameters and $\psi (\xi )$ holds, then f can be chosen so that $\psi (f(i))$ holds for all $i \in I$ . The representation makes possible a coding of the external subsets of $V \cap \mathbf {S}_w$ by standard sets, and the coding process can be continued to the putative higher levels of the external cumulative hierarchy. This process is enabled by the principle of Standardization.

Definition 3.6. Let $\phi (v)$ be a formula in the $\operatorname {\mathrm {st}}$ - $\in $ -language, with arbitrary parameters. We let

$$ \begin{align*}{}^{\operatorname{\mathrm{st}}} \{ x \in A \,\mid\, \phi (x) \}\end{align*} $$

denote the standard set S such that $\forall ^{\operatorname {\mathrm {st}}} x\,(x \in S \longleftrightarrow x \in A \,\wedge \,\phi (x))$ .

The principle of Standardization postulates the existence of this set and Transfer guarantees its uniqueness. Also by Transfer, if $\psi $ is any $\in $ -formula with standard parameters and $\forall ^{\operatorname {\mathrm {st}}} x \in A\; (\phi (x) \rightarrow \psi (x))$ , then $\forall x \in S\; \psi (x)$ .

In particular, if $\phi (u.v)$ is a formula in the $\operatorname {\mathrm {st}}$ - $\in $ -language with arbitrary parameters, A, B are standard, and for every standard $x \in A$ there is a unique standard $y \in B$ such that $\phi (x, y)$ , then there is a unique standard function $F: A \to B$ such that $\forall ^{\operatorname {\mathrm {st}}} x \in A\;\phi (x, F(x))$ holds. It suffices to let $F = {}^{\operatorname {\mathrm {st}}}\{ \langle x, y \rangle \in A \times B \,\mid \, \phi (x,y) \}$ .

Definition 3.7. The $w,V$ -code of an external set $\mathbf {X}\subseteq V \cap \mathbf {S}_w$ is the standard set

$$ \begin{align*}\Psi_{w,V} (\mathbf{X}) =X = {}^{\operatorname{\mathrm{st}}}\{ f \in V^I\,\mid\, f(w) \in \mathbf{X} \}.\end{align*} $$

In words, $\Psi _{w,V} (\mathbf {X}) $ is the standard set whose standard elements are precisely the standard $f\in V^I $ with $f(w) \in \mathbf {X}$ .

We omit the subscripts w and/or V when they are understood from the context.

The coding is trivially seen to preserve elementary set-theoretic operations.

The coding preserves infinite unions and intersections as well. An externally countable sequence of external subsets of V can be viewed in $\mathrm {BST}$ as an external subset $\mathbf {X}$ of $\mathbb {N} \times V$ , with for standard $n \in \mathbb {N}$ . Let $\langle X_n \,\mid \, n \in \mathbb {N} \rangle $ be the standard sequence such that $X_n = \Psi (\mathbf {X}_n)$ holds for all standard n (its existence follows from Standardization). Then

$$ \begin{align*} \Psi(\bigcup_{n \in \mathbb{N} \cap \mathbf{S}} \mathbf{X}_n) = \bigcup_{n \in \mathbb{N}} X_n \end{align*} $$

because if $f \in \bigcup _{n \in \mathbb {N}} X_n$ is standard, then the least $n \in \mathbb {N}$ such that $f \in X_n$ is standard. Similarly,

$$ \begin{align*}\Psi(\bigcap_{n \in \mathbb{N} \cap \mathbf{S}} \mathbf{X}_n) = \bigcap_{n \in \mathbb{N}} X_n \end{align*} $$

because if a standard $f \in \Psi (\bigcap _{n \in \mathbb {N} \cap \mathbf {S}} \mathbf {X}_n)$ , then $f \in X_n$ for all standard $n \in \mathbb {N}$ , and hence $f \in \bigcap _{n \in \mathbb {N}} X_n$ by Transfer.

It is convenient to relax the definition of coding so that every standard $S \subseteq V^I$ is a code of some external $\mathbf {X} \subseteq V \cap \mathbf {S}_w$ .

If S codes $\mathbf {X}$ , then $\Psi (\mathbf {X}) = {}^{\operatorname {\mathrm {st}}} \{f \in V^I \,\mid \, f(w) = g(w) \text { for some standard} g \in S\}.$ A code of $\mathbf {X}$ has to contain a representation for each $\xi \in \mathbf {X}$ , but not necessarily all such representations.

We note that coding is independent of the choice of the universal standard set in the following sense: If $\mathbf {X} \subseteq V_1 \cap \mathbf {S}_w$ and $V_1 \subseteq V_2$ , then S codes $\mathbf {X}$ viewed as a subset of $V_1 \cap \mathbf {S}_w$ iff S codes $\mathbf {X}$ viewed as a subset of $V_2 \cap \mathbf {S}_w$ . Hence the exact choice of V is usually of little importance.

For any set x, let $c_x$ be the constant function with value x. The informal identification of x with $c_x$ enables the identification of a standard set A with a code $ {}^{\operatorname {\mathrm {st}}}\{ c_x \,\mid \, x \in A \} = \{ c_x \,\mid \, x \in A \}$ for $A \cap \,\mathbf {S}$ . On the other hand, the standard set $A^I$ is a code for $A \cap \,\mathbf {S}_w$ .

Every standard $S \subseteq V^I$ is a code of the unique external set

Such S codes a subset of an external set $\mathbf {X} $ iff $\forall ^{\operatorname {\mathrm {st}}} f\, (f \in S \rightarrow f(w) \in \mathbf {X})$ iff $S \subseteq \Psi (\mathbf {X}) $ . Consequently, it would make sense to interpret the power set of $\widetilde {X}=\Psi (\mathbf {X}) $ as a code for the “non-existent” external power set $\mathcal {P}^{\operatorname {\mathrm {ext}}} (\mathbf {X})$ of $\mathbf {X}$ , and the standard subsets of $\mathcal {P}(\widetilde {X})$ as codes for the “external subsets of $\mathcal {P}^{\operatorname {\mathrm {ext}}} (\mathbf {X})$ .” Since $\mathrm {BST}$ does not allow collections of external sets, this last remark cannot be made rigorous in it, but one can proceed “as if” such higher-order external sets existed.

Intuitively, there is a hierarchy of external sets built up over $V \cap \mathbf {S}_w$ :

$$ \begin{align*} \boldsymbol{\mathcal{H}}_1 = \mathcal{P}^{\operatorname{\mathrm{ext}}}(V \cap \mathbf{S}_w) \text{ and } \boldsymbol{\mathcal{H}}_{n+1} = \mathcal{P}^{\operatorname{\mathrm{ext}}} (\boldsymbol{\mathcal{H}}_n) \text{ for standard } n \in \mathbb {N}\end{align*} $$

(we stop here to avoid further complications). This hierarchy cannot be formalized in $\mathrm {BST}$ . But the corresponding hierarchy over $V^I$ is well-defined:

$$ \begin{align*} H_1 = \mathcal{P}(V^I) \text{ and } H_{n+1} = \mathcal{P} (H_n) \text{ for standard } n \in \mathbb{N}. \end{align*} $$

In $\mathrm {BST}$ one can work legitimately in the latter hierarchy while keeping in mind that the coding establishes a (many-one) correspondence between $\boldsymbol {\mathcal {H}}_n$ and $H_n \cap \mathbf {S}$ . Both hierarchies and their relationship could be formalized in $\mathrm {HST}$ , a conservative extension of $\mathrm {BST}$ to a theory that encompasses also external sets [Reference Hrbacek9, Reference Kanovei and Reeken14].

Note that the coding process treats elements $\xi $ of $V \cap \mathbf {S}_w$ as individuals; they are not coded by $\Psi $ . Thus $\xi $ is represented by any f with $f(w) = \xi $ , but $\Psi (\xi )$ is undefined. The set $\{\xi \} \subseteq V\cap \mathbf {S}_w$ is coded by $\{f\}$ , even when $ \{\xi \} \in V\cap \mathbf {S}_w$ . In particular, if $\mathbf {R}$ is a binary relation on $V \cap \mathbf {S}_w$ , then the code for $\mathbf {R}$ is the standard binary relation

$$ \begin{align*}\Psi (\mathbf{R}) =R = \,{}^{\operatorname{\mathrm{st}}}\{ \langle f , g \rangle \,\mid\, f, g \in V^I \,\wedge\, \langle f(w), g(w)\rangle \in \mathbf{X} \}.\end{align*} $$

Similarly for functions and relations of higher arity.

Another variant of coding represents each $\xi \in \mathbf {X}$ by a single object.

Definition 3.10. Let $f \in V^I$ be standard. We let

$$ \begin{align*}f_{w,V} = f_w ={}^{\operatorname{\mathrm{st}}}\{g \in V^I\,\mid\, g(w) = f(w) \}.\end{align*} $$

We define standard sets $V^I/w = {}^{\operatorname {\mathrm {st}}}\{ f_w \,\mid \, f \in V^I\}$ and, for $\mathbf {X} \subseteq V \cap \mathbf {S}_w,$

$$ \begin{align*} \widetilde{\Psi}_w (\mathbf{X}) &= {}^{\operatorname{\mathrm{st}}} \{ f_w \in V^I/w \,\mid \, f(w) \in \mathbf{X}\} \\ &= {}^{\operatorname{\mathrm{st}}} \{ F \in V^I/w \,\mid\, \exists^{\operatorname{\mathrm{st}}} f \in V^I\, (F = f_w \,\wedge\, f(w) \in \mathbf{X}) \}. \end{align*} $$

The coding $\widetilde {\Psi }_w$ is one–one. For standard $X \subseteq V^I/w $ we let

One can obtain $\widetilde {\Psi }_w (\mathbf {X}) $ from $\Psi _w (\mathbf {X}) $ and vice versa:

$$ \begin{align*} \widetilde{\Psi}_w (\mathbf{X}) = {}^{\operatorname{\mathrm{st}}} \{ f_w \,\mid\, f \in \Psi_w(\mathbf{X})\} \text{ and } \Psi_w(\mathbf{X}) = {}^{\operatorname{\mathrm{st}}} \{ f \in V^I \,\mid\, f_w \in \widetilde{\Psi}_w (\mathbf{X}) \}. \end{align*} $$

Proposition 3.8 and the subsequent paragraph hold with $\Psi $ replaced by $\widetilde {\Psi }$ .

We define the hierarchy $\widetilde {H}_1 = \mathcal {P}(V^I/w)$ and $\widetilde {H}_{n+1} = \mathcal {P} (\widetilde {H}_n)$ for standard $n \in \mathbb {N}$ . The coding $\widetilde {\Psi }_w$ maps $\boldsymbol {\mathcal {H}}_1$ onto $\widetilde {H}_1 \cap \mathbf {S}$ . It extends informally to higher levels by $\widetilde {\Psi }_w(\mathbf {X}) = {}^{\operatorname {\mathrm {st}}} \{ \widetilde {\Psi }_w(\mathbf {Y}) \,\mid \, \mathbf {Y} \in \mathbf {X}\}$ and provides a one–one correspondence between $\boldsymbol {\mathcal {H}}_n$ and $\widetilde {H}_n \cap \mathbf {S}$ .

4.1 Nonstandard hulls of standard metric spaces

Let $\mathbb {R}$ be the field of real numbers and let $(M, d)$ be a standard metric space, so that M is a standard set and the distance function is a standard mapping $d: M \times M \to \mathbb {R}$ . A point $x \in M$ is finite if $d(x,p)$ is limited for some (equivalently, for all) standard $p \in M$ .

Fix an unlimited integer $w \in \mathbb {N}$ . We define the standard set $B_w$ by

$$ \begin{align*}B_w =\, {}^{\operatorname{\mathrm{st}}} \{ f \in M^{\mathbb{N}} \,\mid\, f(w) \text{ is a finite point of }M\}.\end{align*} $$

The standard relation $E_w$ on $B_w$ is defined by

$$ \begin{align*}E_w = \, {}^{\operatorname{\mathrm{st}}} \{ \langle f, g \rangle \in B _w\times B_w \,\mid\, d(f(w) , g(w))\simeq 0\}.\end{align*} $$

Clearly $E_w $ is reflexive, symmetric, and transitive on standard elements of $B_w$ ; it follows by Transfer that $E_w$ is an equivalence relation on $B_w$ . We denote the equivalence class of f modulo $E_w$ by $f_{E_w}$ .

By Standardization, there is a standard function $D_w: B_w \times B_w \to \mathbb {R}$ determined by the requirement that $D_w ( f, g ) = \operatorname {\mathrm {sh}} ( d (f(w) , g(w) ))$ for all standard $f, g\in B_w$ .

For standard $f, g\in B_w$ the distance

$$ \begin{align*}d (f(w) , g(w) )) \le d(f(w), f(0)) + d(f(0), g(0)) + d(g(0), g(w))\end{align*} $$

is limited, so $\operatorname {\mathrm {sh}} ( d (f(w) , g(w) )) \in \mathbb {R}$ .

For standard $f,g,h \in B_w$ clearly $D_w ( f, g ) = D_w ( g,f ) $ ,

$$ \begin{align*}D_w ( f, h ) \le D_w ( f, g ) + D_w ( g, h) , \text{ and } D_w ( f, g ) = 0 \text{ iff } \langle f, g \rangle \in E_w.\end{align*} $$

Also, for standard $f,g,f',g' \in B_w$ we have

$$ \begin{align*}\bigl( \langle f, f'\rangle \in E_w \,\wedge \, \langle g, g' \rangle \in E_w \bigr)\rightarrow D_w(f,g) = D _w(f', g'). \end{align*} $$

By Transfer these properties hold for all $f,g ,h, f',g' \in B_w$ .

We observe that, for the natural choice $V = M \cup \mathbb {R}$ , $B_w$ is a code for the external set

$E_w$ is nothing but a code for the external equivalence relation

and $D_w$ is a code for the external function $\mathbf {D}_w: \mathbf {B}_w \times \mathbf {B}_w \to \mathbb {R} \cap \mathbf {S}$ defined by

.

The construction of the nonstandard hull of $(M,d)$ by the usual external method would form the quotient space $\mathbf {B} _w/\mathbf {E}_w$ of $\mathbf {B}_w$ modulo $\mathbf {E}_w$ . This step cannot be carried out in $\mathrm {BST}$ . For $x \in \mathbf {B}_w$ , the equivalence class of x modulo $\mathbf {E}_w $ is , but the collection of the classes $\mathbf {M}_w(x)$ for all $x \in \mathbf {B}_w$ is not supported by $\mathrm {BST}$ . However, for standard $f \in B_w$ , the set $f_{E_w}$ is a code of $\mathbf {M}_w(x)$ when $f(w) = x$ . Therefore $\mathbf {B}_w /\mathbf {E}_w$ can be replaced by $B_w/E_w$ , which is a standard set in $\mathrm {BST}$ . The standard elements of $B_w/E_w$ are precisely the codes of the monads $\mathbf {M}_w(x)$ for $x \in \mathbf {B}_w$ . Hence $B_w/E_w$ is a code (as discussed in Section 3.2) of the “non-existent” (in $\mathrm {BST}$ ) quotient space $\mathbf {B} _w/\mathbf {E}_w$ .

We stress that while external sets serve as a motivation for our constructions, they are not actually used in them; the constructions deal only with sets of $\mathrm {BST}$ . The same applies to the rest of this section.

We let $\widehat {M}_w = B_w / E_w $ be the standard quotient space of $B_w$ modulo $E_w$ . From now on we often omit the subscript w when it is understood from the context.

The function $D $ factors by E to a (standard) function $\widehat {D} = D / E$ on $\widehat {M}$ , defined by $\widehat {D} ( f_E, g_E ) = D(f, g)$ (as shown above, the value of $\widehat {D}$ is independent of the choice of representatives f, g). It is clear that $\widehat {D}$ is a metric on $\widehat {M} $ .

The embedding c of M into $\widehat {M}$ is via constant functions: for $x \in M$ , $c(x) = (c_x)_E$ where $c_x$ is the constant function on $\mathbb {N}$ with value x. Trivially, the embedding c preserves the metrics. We identify M with its image in $\widehat {M}$ under this embedding.

We emphasize that the structure $(\widehat {M} , \widehat {D} \,)$ depends on the choice of the parameter w, an unlimited integer. A metric space has a unique completion up to isometry, but it may have many non-isometric nonstandard hulls.

Example 4.1. Let $M = \mathbb {Q}$ be the set of rationals and $d(x,y) = |x-y|$ be the usual metric on $\mathbb {Q}$ . We fix an unlimited $w\in \mathbb {N}$ . We note that, for standard f,

$$ \begin{align*}f \in B_w \longleftrightarrow f \in \mathbb{Q}^{\mathbb{N}} \,\wedge\, f(w) \text{ is limited},\end{align*} $$

and for standard $f,g \in B_w$

$$ \begin{align*}\langle f, g \rangle &\in E_w \longleftrightarrow f(w) \simeq g(w) \longleftrightarrow f(w),\\ g(w) &\in \mathbf{M}(a) \text{ for } a = \operatorname{\mathrm{sh}}(f(w)) = \operatorname{\mathrm{sh}}(g(w)).\end{align*} $$

By Standardization, there is a standard function $\Gamma : B_w/E_w \to \mathbb {R}$ such that, for standard f, $\Gamma (f/E ) = \operatorname {\mathrm {sh}}(f(w)).$ It is easy to verify that $\Gamma \upharpoonright (B_w/E_w) \cap \mathbf {S}$ is a one–one mapping of $\widehat {M}_w \cap \mathbf {S}$ onto $\mathbb {R} \cap \mathbf {S}$ that preserves the metrics:

$$ \begin{align*}\widehat{D}(f/E, g/E) &= D(f,g) = \operatorname{\mathrm{sh}}(|f(w) - g(w)|)\\ & = |\operatorname{\mathrm{sh}}(f(w)) - \operatorname{\mathrm{sh}}(g(w))| = |\Gamma(f) - \Gamma(g)|.\end{align*} $$

Moreover, $\Gamma ((c_x)_E) = x$ for any standard $x \in \mathbb {Q}$ . By Transfer, $\Gamma $ is an isometric isomorphism of $\widehat {M}_w$ and $\mathbb {R}$ which is the identity on $\mathbb {Q}$ . In particular, the nonstandard hull is independent of the choice of w.

Example 4.2. Let $M= \mathbb {N}$ and let d be the discrete metric on M; i.e., $d(x,z) = 1$ for all $x,z \in M$ , $x \neq z$ . As all points of M are finite with respect to this metric, we have $B_w = \mathbb {N}^{\mathbb {N}}$ . Also

$$ \begin{align*}E_w = {}^{\operatorname{\mathrm{st}}}\{ \langle f, g \rangle \in \mathbb{N}^{\mathbb{N}} \times \mathbb{N}^{\mathbb{N}} \,\mid\, f(w) = g(w)\}\end{align*} $$

and, for standard $f \in \mathbb {N} ^{\mathbb {N}}$ , $f/E = {}^{\operatorname {\mathrm {st}}} \{ g \in \mathbb {N}^{\mathbb {N}} \,\mid \, g(w) = f(w)\} = f_w$ . The space M is identified with a subset of $\widehat {M}_w$ via $\Gamma : (c_x)_w \mapsto x$ .

In Remark 5.5 we establish that $\widehat {M}_w =B_w/E_w = \mathbb {N}^{\mathbb {N}}/w$ is exactly the ultrapower of M by $U_w$ , an ultrafilter over $\mathbb {N}$ generated by w; in particular, it has the cardinality of the continuum. Also, $\widehat {D}$ is the discrete metric on $\widehat {M}_w$ . By replacing $\mathbb {N}$ with I as in Remark 4.6 one can obtain nonstandard hulls of arbitrarily large cardinality.

Example 4.3. Approachable points play an important role in the study of nonstandard hulls. We define the concept as follows.

Let $(M, d)$ be a standard metric space. A point $x \in M$ is approachable if for every standard $\epsilon> 0$ there is a standard $a \in M$ such that $d(x, a) \le \epsilon $ .

The approachable points in $M \cap \mathbf {S}_w$ become exactly the standard points of the closure of M in its nonstandard hull $\widehat {M}_w$ . Indeed, for standard $f \in B_w$ with $f(w) = x \in M$ we have

$$ \begin{align*} x \text{ is approachable } \longleftrightarrow \forall^{\operatorname{\mathrm{st}}} \epsilon> 0 \; \exists^{\operatorname{\mathrm{st}}} a \in M\; (d(x,a) \le \epsilon) \longleftrightarrow \end{align*} $$

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} \epsilon> 0 \; \exists^{\operatorname{\mathrm{st}}} a \in M\; (\widehat{D}(f/E,\,a) \le \epsilon) \longleftrightarrow \forall \epsilon > 0 \; \exists a \in M\; (\widehat{D}(f/E,\,a) \le \epsilon),\end{align*} $$

where the last step is by Transfer. Thus in Example 4.1 all finite $x \in \mathbb {Q}$ are approachable and consequently all standard points in $\mathbb {R}$ are in the closure of $M = \mathbb {Q}$ . By Transfer, this is true for all points in $\mathbb {R}$ . In Example 4.2 all nonstandard points of $\mathbb {N}$ are inapproachable and therefore M is closed in $\widehat {M}_w$ .

4.2 Completeness of the nonstandard hull

This subsection illustrates how one can work with the nonstandard hull as constructed in Section 4.1.

Proof Let $\langle F_n \, \mid \, n \in \mathbb {N} \rangle $ be a standard Cauchy sequence in $(\widehat {M} , \widehat {D}\, )$ ; we prove that it converges to some $F \in \widehat {M}$ .

Using the Axiom of Countable Choice we obtain a standard sequence $\langle f_n \, \mid \, n \in \mathbb {N} \rangle $ such that $F_n = (f_n)_E$ for all $n \in \mathbb {N}$ .

For $k \in \mathbb {N}$ let $n_k$ be the least element of $\mathbb {N}$ greater than or equal to k such that

$$ \begin{align*}\forall m,n \; \left( n_k \le n\le m \rightarrow \widehat{D}(F_n, F_m) <\tfrac{1}{ k+1}\right);\end{align*} $$

note that the sequence $\langle n_k \,\mid \, k \in \mathbb {N} \rangle $ is standard. From the definition of $\widehat {D}$ we obtain, for standard k,

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} m,n \; \left( n_k \le n \le m \rightarrow d(f_n(w), f_m(w)) < \tfrac{1}{ k+1}\right) .\end{align*} $$

Hence $\forall ^{\operatorname {\mathrm {st}}} k \;\exists ^{\operatorname {\mathrm {st}}} m \; \forall \ell \le k$

$$ \begin{align*}\left[\, n_{\ell} \le m \,\wedge\, \forall n \; \left( n_{\ell} \le n \le m \rightarrow d(f_n(w), f_m(w)) < \tfrac{1}{ \ell+1} \right)\right].\end{align*} $$

By Countable Idealization into w-standard sets we get

$$ \begin{align*}\exists^{\operatorname{\mathrm{st}}_{w}} m \; \forall^{\operatorname{\mathrm{st}}} k\,\left[\, n_{k} \le m \,\wedge\, \forall n \; \left( n_{k} \le n \le m \rightarrow d(f_n(w), f_m(w)) < \tfrac{1}{ k+1} \right)\right].\end{align*} $$

Fix such an m; clearly it is unlimited. Consider the standard point $p = f_{n_0}(0)$ . We have $d(p, f_m(w)) \le d(f_{n_0}(0), f_{n_0}(w) ) + d(f_{n_0}(w), f_m(w))$ . The first term on the right side of the inequality is limited because $f_{n_0} $ is standard and $f_{n_0} \in B$ , and the second term is $< 1$ (take $k = 0$ ). Hence $f_m(w)$ is a finite element of M.

We note that $f_m(w)$ is w-standard; hence there is a standard function $f{\kern-1.5pt}\in{\kern-1.5pt} M^{\mathbb {N}}$ such that $f(w) {\kern-1.5pt}={\kern-1.5pt} f_m(w) $ . It follows that $f {\kern-1.5pt}\in{\kern-1.5pt} B$ . We let $F {\kern-1.5pt}={\kern-1.5pt} f_E {\kern-1.5pt}\in{\kern-1.5pt} \widehat {M}$ .

We have that, for all standard k,

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} n \; \left( n_{k} \le n \rightarrow d(f_n(w), f(w)) < \tfrac{1}{ k+1} \right),\end{align*} $$

and hence

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} n \; \left( n_{k} \le n \rightarrow \widehat{D}(F_n, F) \le \tfrac{1}{ k+1} \right).\end{align*} $$

This shows that the sequence $\langle F_n \, \mid \, n \in \mathbb {N} \rangle $ converges to F.

The proof of Theorem 4.2 goes through for any infinite set I in place of $\mathbb {N}$ , as long as $w \in I$ is good.

4.3 Nonstandard hulls of standard uniform spaces

We generalize the construction of nonstandard hulls in $\mathrm {BST}$ to uniform spaces.

Let $(M, \Delta )$ be a standard uniform space. That is, M is a standard set and $\Delta $ is a standard family of pseudo-metrics on M which endows M with a Hausdorff uniform structure. In a uniform space, $x \in M$ is finite if for all standard $d \in \Delta $ , $d(x,p)$ is limited for some (equivalently: for all) standard $p \in M$ . Points x and $ y $ are infinitely close if $d(x,y) \simeq 0$ for all standard $d \in \Delta $ .

We fix a standard infinite set I and a $w \in I$ so that Idealization into w-standard sets over standard sets of cardinality $\le \kappa $ holds for $\kappa = \max \{ |\Delta |, \aleph _0\}$ (see Proposition 3.5). A construction of the nonstandard hull of $(M, \Delta )$ can now proceed much as in Section 4.1. We omit the subscripts indicating its dependence on w.

We let $B = \, {}^{\operatorname {\mathrm {st}}} \{ f \in M^I \,\mid \, f(w) \text { is a finite point of }M\}$ and

$$ \begin{align*} E = \, {}^{\operatorname{\mathrm{st}}} \{ \langle f, g \rangle \in B \times B \,\mid\, d(f(w), g(w)) \simeq 0 \text{ for all standard } d \in \Delta \}. \end{align*} $$

For each standard $d \in \Delta $ the standard function D on $B \times B$ , as well as $\widehat {M}$ , $\widehat {D}$ and c, are defined as in Section 4.1. Each $\widehat {D}$ is a pseudo-metric on $\widehat {M}$ . We let $\widehat {\Delta } =\, {}^{\operatorname {\mathrm {st}}} \{\widehat {D} \,\mid \, d \in \Delta \cap \mathbf {S}\}$ .

Proof The proof follows the lines of the proof of Theorem 4.7; the main difference is that Cauchy sequences have to be replaced by Cauchy nets.

Let $\langle \Lambda , \le \rangle $ be a standard directed set and $\langle F_{\lambda } \,\mid \, \lambda \in \Lambda \rangle $ a standard Cauchy net indexed by $\Lambda $ . Using the Axiom of Choice one obtains a standard net $\langle f_{\lambda } \,\mid \, \lambda \in \Lambda \rangle $ such that $f_{\lambda } \in F_{\lambda }$ holds for all $\lambda $ ; then $\langle f_{\lambda }(w) \,\mid \, \lambda \in \Lambda \rangle $ is a w-standard net.

The Cauchy property implies that for every standard $d\in \Delta $ there is a standard sequence $\langle \lambda _k^d \,\mid \,k \in \mathbb {N} \rangle $ of elements of $\Lambda $ such that $\lambda _k^d \le \lambda _{k+1}^d $ holds for all k and

$$ \begin{align*}\forall k\,\forall \lambda, \mu \in \Lambda \; \left( \lambda_k^d \le \lambda \le \mu \rightarrow \widehat{d}(F_{\lambda}, F_{\mu}) <\tfrac{1}{ k+1}\right) .\end{align*} $$

Hence

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} k\,\forall^{\operatorname{\mathrm{st}}} \lambda, \mu \in \Lambda \; \left( \lambda_k^d \le \lambda \le \mu \rightarrow d(f_{\lambda}(w), f_{\mu}(w)) <\tfrac{1}{k+1}\right).\end{align*} $$

We conclude that for every standard $k \in \mathbb {N}$ , every standard finite $\Delta _0 \subseteq \Delta $ and every standard finite $\Lambda _0 \subseteq \Lambda $ there is a standard $\mu \in \Lambda $ such that for all $\ell \le k$ , all $d \in \Delta _0$ and all $\lambda \in \Lambda _0$

$$ \begin{align*}\lambda_{\ell}^d \le \mu \,\wedge\, \left( \lambda_{\ell}^d \le \lambda \rightarrow d(f_{\lambda}(w), f_{\mu}(w)) < \tfrac{1}{\ell+1} \right).\end{align*} $$

By Idealization into w-standard sets over sets of cardinality $\le \kappa = \max \{ |\Delta |, \aleph _0\}$ we get a w-standard $\mu \in \Lambda $ such that for all standard $ k$ , all standard $ d \in \Delta $ and all standard $\lambda \in \Lambda $

$$ \begin{align*}\lambda_k^d \le \mu \,\wedge\, \left( \lambda_k^d \le \lambda \rightarrow d(f_{\lambda}(w), f_{\mu}(w)) < \tfrac{1}{k+1} \right).\end{align*} $$

Fix such a $\mu $ ; as in Section 4.7 we have that $d(p, f_{\mu }(w))$ is limited for every standard $d \in \Delta $ , i.e., $f_{\mu }(w)$ is a finite point of M.

By Representability there is a standard function $f \in M^I$ such that $f(w) = f_{\mu }(w)$ . It follows that $f \in B$ . Let $F = f/E \in \widehat {M}$ .

We have that, for all standard $d \in \Delta $ and $k \in \mathbb {N}$ ,

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} \lambda \in \Lambda \; \left( \lambda_{k}^d \le \lambda \rightarrow d(f_{\lambda}(w), f(w)) < \tfrac{1}{k+1} \right) ,\end{align*} $$

and hence

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} \lambda \in \Lambda\; \left( \lambda_{k}^d \le \lambda \rightarrow \widehat{D}(F_{\lambda}, F) \le \tfrac{1}{k+1}\right).\end{align*} $$

This shows that the net $\langle F_{\lambda } \, \mid \, \lambda \in \Lambda \rangle $ converges to F.

4.4 Internal normed vector spaces

Under many circumstances the type of construction carried out in Sections 4.1 and 4.3 generalizes to internal structures. We illustrate it in the case of normed vector spaces.

Let M be an internal normed vector space over $\mathbb {R}$ . This means that M, the operations of addition $+$ on $M \times M$ and scalar multiplication $\cdot $ on $\mathbb {R} \times M$ , and the $\mathbb {R}$ -valued norm $ \|. \| $ on M, are (internal) sets and satisfy the usual properties. In order to be able to apply our coding technique we fix a standard set I and a good $w \in I$ so that the set M, the above operations, and the norm belong to $\mathbf {S}_w$ . Other parameters relevant to a particular investigation can also be made to belong to $\mathbf {S}_w$ . By Transfer in $(\mathbf {V}, \mathbf {S}_w, \in )$ , the properties of these objects that are expressible by $\in $ -formulas continue to hold in $\mathbf {S}_w$ , so we can carry out the desired construction “over $\mathbf {S}_w$ ” rather than “over $\mathbf {V}$ .”

We first describe the external construction. Let

It is clear that $\mathbf {B}_w$ is an external vector space over the external field $\mathbb {R} \cap \mathbf {S}$ and $\mathbf {E}_w$ is its subspace. Define an external equivalence relation $\approx $ on $\mathbf {B}_w$ by $x \approx y \longleftrightarrow x - y \in \mathbf {E}_w \longleftrightarrow \| x-y \| \simeq 0$ . Obviously, for $x_1, x_2, y_1, y_2 \in \mathbf {B}_w$ and $c \in \mathbb {R} \cap \mathbf {S}$ we have $x_1 \approx y_1 \,\wedge \, x_2 \approx y_2 \rightarrow x_1 + x_2 \approx y_1 + y_2$ , $c\cdot x_1 \approx c\cdot y_1$ and $\|x_1\| \simeq \|y_1\|$ . If external collections of external sets were available in $\mathrm {BST}$ , one could now form the quotient space $\mathbf {B}_w/\mathbf {E}_w$ with the norm $\|x_{\mathbf {E}_w}\| = \operatorname {\mathrm {sh}}(\|x\|)$ , which would then be an external normed vector space over the external field $\mathbb {R} \cap \mathbf {S}$ . This construction is of course not possible in $\mathrm {BST}$ directly, but the coding introduced in Section 3.2 enables us to carry it out and produce a standard normed metric space which, when viewed from the standard point of view, is (externally) isomorphic to $\mathbf {B}_w/\mathbf {E}_w$ .

It follows from Representability that there is a standard function $\langle (M_i, \,+_i, \, \cdot _i) \,\mid \, i \in I \rangle $ such that $(M_w, \,+_w, \, \cdot _w) = (M, \,+, \, \cdot )$ and, for all $i \in I$ , $(M_i, \,+_i, \, \cdot _i)$ is a vector space over $\mathbb {R}$ . Let

$$ \begin{align*}B_w &= {}^{\operatorname{\mathrm{st}}} \{f \in \Pi_{i \in I} M_i \,\mid\, \|f(w)\| \text{ is limited} \} ;\\E_w &= {}^{\operatorname{\mathrm{st}}} \{f \in \Pi_{i \in I} M_i \,\mid\, \|f(w)\| \simeq 0\}. \end{align*} $$

Addition and scalar multiplication on $B_w$ are defined pointwise:

$$ \begin{align*}(f+ g)(i) = f(i) +_i g(i) \qquad\text{and} \qquad (c \cdot f)(i) = c \cdot_i f(i)\end{align*} $$

for $f, g \in B_w$ , $c \in \mathbb {R}$ , and all $i \in I$ . By Standardization, there is a standard function $\|.\|$ such that for all standard $f \in B_w$ we have $\|f\| = \operatorname {\mathrm {sh}}(\|f(w)\|)$ .

It is routine to verify that $B_w$ is a standard vector space over $\mathbb {R}$ , $\|.\|$ is a pseudo-norm on $B_w$ , and $f \in E_w \longleftrightarrow \|f\| = 0$ . The quotient $\widehat {E}_w = B_w/E_w$ is thus a well-defined standard normed vector space. The proof given in Section 4.2 shows that $\widehat {E}_w = B_w/E_w$ is complete.

4.5 Loeb measures

Let $(\Omega , \mathcal {A}, \mu )$ be an internal finitely additive measure space, with $\Omega \subseteq O$ for a standard set O. As discussed in Section 6.3, attempts to construct its Loeb extension “over $\mathbf {V}$ ” are only partially successful in $\mathrm {BST}$ . We fix a standard set I and a good $w\in I$ so that $\Omega , \mathcal {A} , \mu $ are w-standard. By Transfer, $(\Omega , \mathcal {A}, \mu )$ is an internal finitely additive measure space in the sense of $\mathbf {S}_w$ , and we construct the Loeb extension “over $\mathbf {S}_w$ .” We usually do not indicate the dependence on the choice of w. In this example it is convenient to employ the variant of coding from Definition 3.10.

For every $X \in \mathcal {A}\cap \mathbf {S}_w$ let

(1) $$ \begin{align} [X] =\, {}^{\operatorname{\mathrm{st}}}\{ f_w \,\mid\, f \in O^I \,\wedge\, f(w) \in X \cap \mathbf{S}_w \}. \end{align} $$

If $X, X_1, X_2 \in \mathcal {A}\,\cap \,\mathbf {S}_w$ , then the equivalences $ f_w \in [X_1 \cap X_2] $ iff $f_w \in [X_1]\,\wedge \, f_w \in [X_2]$ and $f_w \in [\Omega \setminus X]$ iff $f_w\in [\Omega ] \,\wedge \,f_w \notin [X]$ hold for all standard $f \in O^I$ . By Transfer, $ [X_1 \cap X_2] = [X_1]\cap [X_2]$ and $ [\Omega \setminus X]= [\Omega ] \setminus [X]$ . Let

$$ \begin{align*}\mathcal{B} =\, {}^{\operatorname{\mathrm{st}}} \{ A \in \mathcal{P}(O^I)\,\,\mid\, A = [X] \text{ for some } X \in \mathcal{A} \cap \mathbf{S}_w\} .\end{align*} $$

Using Transfer again, it follows that $\mathcal {B}$ is a standard algebra of subsets of $[\Omega ]$ . We note that $X_1, X_2 \in \mathcal {A} \cap \mathbf {S}_w$ , $X_1 \ne X_2$ , implies $[X_1] \ne [X_2]$ , so for standard $A \in \mathcal {B}$ there is a unique $X \in \mathcal {A} \cap \mathbf {S}_w$ with $A = [X]$ .

A standard finitely additive measure m on the algebra $\mathcal {B}$ with values in the interval $[0, +\infty ]$ is determined by the requirement that for standard $A = [X] \in \mathcal {B}$

$$ \begin{align*}m(A) = \operatorname{\mathrm{sh}}(\mu(X)) .\end{align*} $$

The measure space $([\Omega ], \mathcal {B}, m)$ satisfies Carathéodory’s condition.

We conclude that m can be extended to a $\sigma $ -additive measure $\overline {m}$ with values in $[0, +\infty ]$ on the $\sigma $ -algebra $\overline {\mathcal {B}}$ generated by $\mathcal {B}$ . The measure-theoretic completion of $([\Omega ], \overline {\mathcal {B}}, \overline {m})$ is the desired Loeb measure space; we denote it $([\Omega ], \widehat {\mathcal {B}}, \widehat {m})$ (of course, it depends on the choice of w). Instead of an appeal to the Carathéodory’s theorem, a direct proof along the lines of [Reference Hrbacek10] can be given; see also [Reference Albeverio, Høegh-Krohn, Fenstad and Lindstrøm1, Remark 3.1.5] and the references therein.

4.6 Lebesgue measure from Loeb measure

As is well known, the Lebesgue measure can be obtained from a suitable Loeb measure.Footnote ⁶ We give the gist of the argument in our framework, for the interval $[0, 1]$ . More details can be found in [Reference Albeverio, Høegh-Krohn, Fenstad and Lindstrøm1] (using a model-theoretic approach).

For $n \in \mathbb {N}$ let $\mathcal {T}_n = \{ i /n \,\mid \, 0 \le i \le n\}$ . Fix a nonstandard integer $w \in \mathbb {N}$ , and let $\mathcal {T} = \mathcal {T}_{w}$ (in model-theoretic frameworks the equivalent of $\mathcal {T}$ is called “hyperfinite time line”). Let $O = [0,1]$ , $\Omega = \mathcal {T}$ , $\mathcal {A} = \mathcal {P}(\mathcal {T})$ and $\mu $ the counting measure on $\mathcal {T}$ , i.e., $\mu (X) = |X|/ |\mathcal {T}|$ for all sets $X \subseteq \mathcal {T}$ . The construction in the preceding subsection, with $I =\mathbb {N}$ , yields the Loeb measure $([\Omega ], \widehat {\mathcal {B}}, \widehat {m})$ .

For every standard $A \subseteq [0,1]$ define

$$ \begin{align*}{}^{\bullet}\!\!A =\, {}^{\operatorname{\mathrm{st}}}\{ f _w\in [\Omega] \,\mid\, f(w) \simeq c \text{ for some standard } c \in A\}.\end{align*} $$

This just means that ${}^{\bullet }\!\!A$ is a w-code for $ \operatorname {\mathrm {sh}}^{-1} (A) \cap \mathcal {T} \cap \mathbf {S}_w $ . Standard elements of the set ${}^{\bullet }\!\!A$ are those $f_w \in [\Omega ]$ whose value “at infinity” (i.e., at w) is infinitely close to a standard real in A.

Let $\mathcal {L} =\, {}^{\operatorname {\mathrm {st}}}\{ A \subseteq [0,1] \,\mid \,{}^{\bullet }\!\!A \in \widehat {\mathcal {B}} \,\}$ and let $\ell $ be the standard function on $\mathcal {L}$ determined by the requirement that $\ell (A) = \widehat {m}({}^{\bullet }\!\!A) $ for all standard $A \in \mathcal {L}$ .

4.7 Neutrices and external numbers

This is another application of nonstandard analysis that extensively uses external sets; see [Reference Dinis and van den Berg6].

A neutrix is a convex additive subgroup of $\mathbb {R}$ . With the exception of $\{0\}$ and $\mathbb {R}$ , neutrices are externals sets; the monad $\mathbf {M}(0)$ and the galaxy $\mathbf {G}(0)$ are nontrivial examples. An external number is an algebraic sum of a real number and a neutrix. Addition and multiplication of external numbers are defined by the Minkowski operations. Let $\mathcal {N}$ denote the collection of all neutrices and $\mathcal {E}$ denote the collection of all external numbers. Even in $\mathrm {HST}$ , $\mathcal {N}$ and $\mathcal {E}$ are (definable) proper classes of external sets; they are “too large” to be external sets (see [Reference Kanovei and Reeken13]).

This difficulty can be remedied by relativizing these concepts to $\mathbf {S}_w$ (for good w). In fact, and . The collections $\mathcal {N}_w$ and $\mathcal {E}_w $ are external sets (of external sets) in $\mathrm {HST}$ .

As the natural embedding of $\mathbb {R} \cap \mathbf {S}_w$ into $\mathcal {E}_w$ given by $r \mapsto r+\{0\}$ is crucial for applications of these concepts, it may be best for most purposes to avoid coding as much as possible. For example, the study of algebraic properties of operations $+$ and $\times $ on $\mathcal {E}_w$ can be carried out in $\mathrm {BST}$ while viewing external numbers as external subsets of $\mathbb {R} \cap \mathbf {S}_w$ . However, work with external numbers often focuses on the structure $(\mathcal {E}_w , +, \times )$ , its subsets, functions with values in it, and so on. Then one can use the techniques of Section 3.2; see in particular Definition 3.10 with $V = \mathbb {R}$ , to code $\mathcal {N}_w$ and $\mathcal {E}_w$ by standard structures.

First, the external set $\mathbb {R} \cap \mathbf {S}_w$ is coded by the standard set $\mathbb {R}^I/w$ . As shown in Remark 5.5, $\mathbb {R}^I/w= \mathbb {R}^I/U_w = {}^{\ast }\mathbb {R}$ , the hyperreals constructed as the standard ultrapower of $\mathbb {R}$ by the standard ultrafilter $U_w$ generated by w. Let ${}^{\ast }\!\!< $ , ${}^{\ast }\!+ $ , and ${}^{\ast }\!\times $ be the ordering, addition, and multiplication on the hyperreals. The coding provides an external isomorphism between $({}^{\ast }\mathbb {R} \cap \mathbf {S}, {}^{\ast }\!\!<, {}^{\ast }\!+, {}^{\ast }\!\times ) $ and $(\mathbb {R} \cap \mathbf {S}_w ,<, +, \times ) $ , so we can informally identify ${}^{\ast }\mathbb {R} \cap \mathbf {S}$ and $\mathbb {R} \cap \mathbf {S}_w$ . Neutrices and external numbers in the hyperreals $({}^{\ast }\mathbb {R}, {}^{\ast }\!\!<, {}^{\ast }\!+, {}^{\ast }\!\times ) $ can be defined the same way as in $\mathbb {R}$ . External numbers in $\mathbb {R} \cap \mathbf {S}_w$ are coded by the standard external numbers in the hyperreals ${}^{\!\ast } \mathbb {R}$ . The coding preserves algebraic operations on external numbers. The collections $\mathcal {N}_w$ and $\mathcal {E}_w$ are coded respectively by the standard sets N and E of all neutrices and external numbers in ${}^{\!\ast } \mathbb {R}$ .

This approach is admittedly rather awkward. In $\mathrm {HST}$ the universes $\mathbf {S}_w$ can be extended to “external universes” $\mathbf {WF}(\mathbf {S}_w)$ (see [Reference Kanovei and Reeken14, Sections 6.3 and 6.4] for details). Perhaps the most practical way to handle external numbers would be to work with $\mathcal {N}_w$ and $\mathcal {E}_w$ in these universes and in the end code the final results in $\mathrm {BST}$ , if desired.

5.1 Subuniverses $\mathbf {S}_w$ and ultrapowers

The universe $\mathbf {S}_w$ is closely connected to the ultrapower of the standard universe by a standard ultrafilter.

The principle of Standardization implies that there is a standard set $U_w$ such that

(2) $$ \begin{align} \forall^{\operatorname{\mathrm{st}}} X \, ( X \in U_w \longleftrightarrow X \in \mathcal{P}(I) \,\wedge\, w \in X). \end{align} $$

Clearly

1. (i) $\emptyset \notin U_w$ , and for standard $X, Y \in \mathcal {P}(I)$

2. (ii) $X \in U_w \,\wedge \, X \subseteq Y \rightarrow Y \in U_w$ ;

3. (iii) $ X, Y \in U_w \rightarrow X \cap Y \in U_w$ ;

4. (iv) $ X \in U_w \,\vee \, (I\setminus X) \in U_w$ .

By Transfer, (ii)–(iv) hold for all $X, Y \in \mathcal {P}(I)$ , so $U_w$ is an ultrafilter over I. One sees easily that $U_w$ is nonprincipal if and only if w is nonstandard. Conversely, Bounded Idealization of $\mathrm {BST}$ implies that for every standard ultrafilter U over I there are (many) $w \in I$ such that $U = U_w$ .

The ultrapower of the universe of all sets $\mathbf {V}$ by a standard ultrafilter U is defined in the usual way. One defines an equivalence relation $=_U$ on $ \mathbf {V}^I$ by

(3) $$ \begin{align} f =_U g \longleftrightarrow \{ i \in I \,\mid\, f(i) = g(i) \} \in U, \end{align} $$

and a membership relation

(4) $$ \begin{align} f \in_U g \longleftrightarrow \{ i \in I \,\mid\, f(i) \in g(i) \} \in U. \end{align} $$

The usual procedure at this point is to form equivalence classes $f_U$ of functions $f \in \mathbf {V}^I$ modulo $=_U$ , using “Scott’s trick” of taking only the functions of the minimal von Neumann rank to guarantee that the equivalence classes are sets: Let

$$ \begin{align*}f_U = \{g \in \mathbf{V}^I\,\mid\, g =_U f \text{ and } \forall h\in \mathbf{V}^I\, (h =_U f \rightarrow \operatorname{rank} h \ge \operatorname{rank} g)\};\end{align*} $$

see [Reference Jech12, (9.3) and (28.15)]. One lets $\mathbf {V}^I\!/U $ be the class of all $ f_U$ for $ f \in \mathbf {V}^I$ and defines

(5) $$ \begin{align} f_U \in_U g_U \longleftrightarrow f \in_U g. \end{align} $$

The ultrapower of $\mathbf {V}$ by U is the structure $(\mathbf {V}^I\!/U,\, \in _U)$ . The universe $\mathbf {V}$ is embedded into $\mathbf {V}^I\!/U$ via $x \mapsto (c_x)_{U}$ where $c_x$ is the constant function on I with value x.

We note that $f_U$ is standard iff $f_U = g_U $ for some standard $g \in \mathbf {V}^I$ . We assume from now on that whenever the equivalence class $f_U$ is standard, the representative function f is taken to be standard.

The key insight is that the standard elements of the ultrapower of $\mathbf {V}$ by $U_w$ are in equality-and-membership-preserving correspondence with w-standard elements of $\mathbf {V}$ . It is expressed by the following proposition, which is an immediate consequence of definitions (3)–(5).

Proposition 5.1. For any standard functions f, $ g \in \mathbf {V}^I$ :

$$ \begin{align*}f =_{U_w} g \longleftrightarrow f_{U_w} = g_{U_w} \longleftrightarrow f(w) = g(w) \quad \text{and}\end{align*} $$

$$ \begin{align*}f \in_{U_w} g \longleftrightarrow f_{U_w} \in_{U_w} g_{U_w} \longleftrightarrow f(w) \in g(w).\end{align*} $$

The correspondence ${\boldsymbol\Phi}_w$ is defined on $\mathbf {S}_w \times (\mathbf {V} ^{I}\!/U_w \,\cap \, \mathbf {S})$ by

$$ \begin{align*}{\boldsymbol\Phi}_w( \xi, f_{U_w}) \longleftrightarrow f(w) = \xi.\end{align*} $$

In this notation, Proposition 5.1 asserts the following.

We note that $(\mathbf {V}^I\!/U_w \,\cap \, \mathbf {S} , \, \in _{U_w} )$ is the ultrapower of the universe in the sense of the standard universe $\mathbf {S}$ . If $\phi (v) $ is an $\in $ -formula such that , then . If $\psi (u,v)$ is an $\in $ -formula such that $F \in _{U_w} G \longleftrightarrow \psi (F,G)$ , then the equivalence $F \in _{U_w} G \longleftrightarrow \psi ^{\operatorname {\mathrm {st}}} (F,G)$ holds for $F,G \in \mathbf {S}$ .

If ${\boldsymbol\Phi}_w (\xi , f_{U_w})$ holds, we write ${\boldsymbol\Phi}_w(\xi ) = f_{U_w}$ . We note that for $x \in \mathbf {S}$ , ${\boldsymbol\Phi}_w(x) = (c_x)_{U_w}$ where $c_x$ is the constant function on I with value x. As is customary, we identify $(c_x)_{U_w}$ with x. This gives a stronger version of Corollary 5.2.

We also note that ${\boldsymbol\Phi}_w(w) = Id_{U_w}$ where $Id(i) = i $ for all $i \in I$ .

Recall that quantifiers with superscript $\operatorname {\mathrm {st}}_w$ range over w-standard sets. If $\phi $ is an $\in $ -formula, $\phi ^{\operatorname {\mathrm {st}}_w}$ is the formula obtained from $\phi $ by relativizing all quantifiers to $\operatorname {\mathrm {st}}_w$ . Łoś’s Theorem for $\in $ -formulas takes the following form:

Lemma 5.4. For standard $f_1,\ldots , f_r$ ,

$$ \begin{align*}\phi^{\operatorname{\mathrm{st}}_w} (f_1(w), \ldots, f_r (w)) \longleftrightarrow \{ i \in I \,\mid\, \phi( f_1(i),\ldots, f_r(i)) \} \in U_w.\end{align*} $$

The structures $(\mathbf {V}^I\!/U_w , \, \in _{U_w} )$ and $(\mathbf {V}^I\!/U_w \,\cap \, \mathbf {S} , \, \in _{U_w} )$ are not models in the sense of model theory because their components are proper classes; hence the satisfaction relation $\vDash $ is not available. Given an $\in $ -formula $\phi $ with parameters from $\mathbf {V}^I\!/U_w $ , we write “ $\phi $ holds in $(\mathbf {V}^I\!/U_w , \, \in _{U_w })$ ” to stand for the formula obtained from $\phi $ by replacing all occurrences of $u \in v$ with $ u\in _{U_w}\! v$ and relativizing all quantifiers to $\mathbf {V}^I\!/U_w $ ; similarly for “ $\phi $ holds in $(\mathbf {V}^I\!/U_w\,\cap \, \mathbf {S} , \, \in _{U_w} )$ .”

Proof of Lemma 5.4

We have

$$ \begin{align*} \phi^{\operatorname{\mathrm{st}}_w} (f_1(w), \ldots, f_r (w)) \longleftrightarrow \end{align*} $$

$$ \begin{align*} \phi((f_1)_{U_w},\ldots,(f_r)_{U_w}) \text{ holds in } (\mathbf{V}^I\!/U_w \cap \mathbf{S} , \, \in_{U_w} ) \quad \text{[by Corollary 5.2] } \end{align*} $$

$$ \begin{align*} \longleftrightarrow \phi((f_1)_{U_w},\ldots,(f_r)_{U_w}) \text{ holds in } (\mathbf{V}^I\!/U_w, \, \in_{U_w} ) \quad \text{[by Transfer]} \end{align*} $$

Remark 5.5. In Section 3.2 we fix a universal standard set V and for standard $f \in V^I$ define $f_{w,V}$ (see Definition 3.10). Clearly

$$ \begin{align*}f_{w,V} &= {}^{\operatorname{\mathrm{st}}}\{ g\in V^I \,\mid\, g(w) = f(w)\}\\ &= {}^{\operatorname{\mathrm{st}}}\{ g\in V^I \,\mid\, g =_{U_w} f\} = \{ g\in V^I \,\mid\, g =_{U_w} f\},\end{align*} $$

where the last step is by Transfer. Thus, again by Transfer, $V^I/w$ is nothing but the ultrapower $V^I/U_w$ .

5.2 Proofs of claims in Section 3.1

Of course, ultrapowers by countably incomplete ultrafilters are the ones of interest in nonstandard analysis.

Proof of Proposition 3.4

Proof of (1)

Assume $\exists x\; \phi (x, p_0)$ where $p_0$ is (wlog. the only) parameter and $\operatorname {\mathrm {st}}_w (p_0)$ . Fix a standard set P such that $p_0 \in P$ (Boundedness). Use $\mathrm {AC}$ to obtain a standard function F on P such that $\forall p \in P\, ( \exists x\, \phi (x,p) \rightarrow \phi ( F(p), p) ).$ Hence $\phi (F(p_0), p_0)$ holds. As $F(p_0) $ is w-standard by Proposition 3.2 (c), we conclude that $\exists ^{\operatorname {\mathrm {st}}_w} x\; \phi (x, p_0)$ . $\Box $

Proof of (2)

It is well-known that every ultrapower by a countably incomplete ultrafilter U is $\omega _1$ -saturated (see [Reference Chang and Keisler4, Theorem 6.1.1]). Hence Countable Idealization in the form

(6) $$ \begin{align} \forall^{\operatorname{\mathrm{st}}} n \in \mathbb{N}\;\exists x\; \forall m \in \mathbb{N}\; (m \le n \;\rightarrow \phi(m,x))\longleftrightarrow \exists x \; \forall^{\operatorname{\mathrm{st}}} n \in \mathbb{N} \; \phi(n,x)\end{align} $$

holds in $(\mathbf {V}^I\!/U_w , \, \mathbf {V}, \, \in _{U_w} )$ . By $\mathrm {BST}$ Transfer, (6) holds in $(\mathbf {V}^I\!/U_w \cap \mathbf {S}, \, \mathbf {S}, \, \in _{U_w} )$ , and, by Corollary 5.3, it holds in $(\mathbf {S}_w, \mathbf {S}, \in )$ . This translates to

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} n \in \mathbb{N}\;\exists^{\operatorname{\mathrm{st}}_w} x\; \forall^{\operatorname{\mathrm{st}}_w} m \in \mathbb{N}\; (m \le n \;\rightarrow \phi^{\operatorname{\mathrm{st}}_w}(m,x))\longleftrightarrow \exists^{\operatorname{\mathrm{st}}_w} x \; \forall^{\operatorname{\mathrm{st}}} n \in \mathbb{N} \; \phi^{\operatorname{\mathrm{st}}_w}(n,x).\end{align*} $$

Using w-Transfer we get the desired form

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}} n \in \mathbb{N}\;\exists x\; \forall m \in \mathbb{N}\; (m \le n \;\rightarrow \phi(m,x))\longleftrightarrow \exists^{\operatorname{\mathrm{st}}_w} x \; \forall^{\operatorname{\mathrm{st}}} n \in \mathbb{N} \; \phi(n,x). \end{align*} $$

Proof of (3)

Recall that $\operatorname {\mathrm {st}}_w(x)$ means that $x = g(w)$ for some standard g defined on I. Let $\phi (v)$ be an $\in $ -formula with standard parameters. Assume $\phi (g(w))$ ; by w-Transfer then $\phi ^{\operatorname {\mathrm {st}}_w}(g(w))$ . From the isomorphism between $(\mathbf {S}_w, \in )$ and $(\mathbf {V}^I/U_w) \cap \mathbf {S}, \,\in _{U_w})$ and Łoś’s Theorem it follows that $ X = \{ i \in I \,\mid \, \phi (g(i))\} \in U_w$ . Pick $i_0 \in X$ and let f be the standard function defined by

$$ \begin{align*}f(i) = g(i) \text{ for } i \in X; \;f(i) = f(i_0) \text{ otherwise.}\end{align*} $$

Then $f(w) = x$ and $\phi (f(i))$ holds for all $i \in I$ .

We prove Proposition 3.5 in the following form.

Proposition 5.8 (Idealization into w-standard sets over sets of cardinality $\le \kappa $ ).

Let $\phi $ be an $\in $ -formula with w-standard parameters. If w is $\kappa ^+$ -good, then for every standard set A of cardinality $\le \kappa $ ,

$$ \begin{align*}\forall^{\operatorname{\mathrm{st}}\! \operatorname{\mathrm{fin}}} a \subseteq A \;\exists y\; \forall x \in a\; \phi(x,y) \longleftrightarrow \exists^{\operatorname{\mathrm{st}}_w} y \; \forall^{\operatorname{\mathrm{st}}} x \in A \; \phi(x,y).\end{align*} $$

For every standard uncountable cardinal $\kappa $ and every z there exist $\kappa ^+$ -good w so that z is w-standard.

Proof of Proposition 5.8

It is well known [Reference Chang and Keisler4, Theorem 6.1.8] that any ultrapower by a countably incomplete $\kappa ^+$ -good ultrafilter U is $\kappa ^+$ -saturated. As in the proof of (2), it follows that Bounded Idealization over sets of cardinality $\le \kappa $ holds in $(\mathbf {S}_w, \mathbf {S}, \in )$ for $\kappa ^+$ -good w.

To also obtain $z \in \mathbf {S}_w$ we use the following fact proved in [Reference Keisler15]:

If U is a countably incomplete $\kappa ^+$ -good ultrafilter over I and V is an ultrafilter over J, then the ultrafilter $U \otimes V$ over $I \times J$ defined by

$$ \begin{align*}X \in U \otimes V \longleftrightarrow \{ i \in I \,\mid\, \{ j \in J \,\mid\, \langle i, j\rangle \in X \} \in V\} \in U\end{align*} $$

is countably incomplete and $\kappa ^+$ -good.

The definition of $\otimes $ implies that for every standard $X \in U \otimes U_z$ there is a standard $i \in I$ such that $\{ j \in J \,\mid \, \langle i, j\rangle \in X \} \in U_z$ , and hence $\langle i, z \rangle \in X$ . From Bounded Idealization one obtains $w \in I$ such that $\langle w, z \rangle \in X$ holds for all standard $X \in U \otimes U_z$ ; in other words, $U_{\langle w,z\rangle } = U \otimes U_z $ . Then $z \in \mathbf {S}_{\langle w,z\rangle }$ and $\langle w,z\rangle $ is $\kappa ^+$ -good, so $\mathbf {S}_{\langle w,z\rangle }$ satisfies Bounded Idealization over sets of cardinality $\le \kappa $ .

Remark 5.9 (More general universes).

The definition of w-standard sets in Section 3.1 can be generalized. Let $w: S \to I$ where $S, I$ are standard sets. We let

It turns out that these universes correspond precisely to the standard limit ultrapowers of the standard universe. The proof is similar to the model-theoretic proof that every internal universe ${}^{\ast }\!V(X)$ is a bounded limit ultrapower of the superstructure $V(X)$ ; see [Reference Chang and Keisler4, Theorems 4.4.19 and 6.4.10]. With suitable modifications, all results described in this paper remain valid for this more general notion of w-standard sets.

6.1 The “full” nonstandard hull

Let $(M, d)$ be a standard metric space. A straightforward attempt to carry out the construction of the nonstandard hull of $(M, d)$ in $\mathrm {BST}$ can start as follows. Let

Let $\mathbf {E}_{\max }$ be the equivalence relation on $\mathbf {B}_{\max }$ defined by

Finally, let the function $\mathbf {D}_{\max }$ with standard real values be defined by

The classes $\mathbf {B}_{\max }, \mathbf {E}_{\max }$ , and $\mathbf {D}_{\max }$ are (definable) external sets. For every $x \in \mathbf {B}_{\max }$ , the equivalence class

is an external set. However, the final step in the construction of the nonstandard hull of $(M,d)$ , to wit, the formation of the quotient space $(\mathbf {B}_{\max }/\mathbf {E}_{\max }, \mathbf {D}_{\max }/\mathbf {E}_{\max })$ , cannot be carried out in $\mathrm {BST}$ (it would require a “class of classes”

).

There are some ways around this difficulty. Perhaps the most straightforward is to forgo the formation of the quotient space and work with the representatives of the equivalence classes (i.e., the elements of $\mathbf {B}_{\max }$ ), and with the congruence $\mathbf {E}_{\max }$ in place of the actual equality. This would be similar to working with fractions rather than the rational numbers. But this way does not produce the nonstandard hull as an actual object of $\mathrm {BST}$ .

The quotient space $(\mathbf {B}_{\max }/\mathbf {E}_{\max }, \mathbf {D}_{\max }/\mathbf {E}_{\max })$ can be formed in $\mathrm {HST}$ (using its axiom of Replacement for $\operatorname {\mathrm {st}}$ - $\in $ -formulas). An interpretation of $\mathrm {HST}$ can be coded in $\mathrm {BST}$ (see [Reference Kanovei and Reeken14, Definition 5.1.2, Theorem 5.1.4, and Corollary 5.1.5]), so in this indirect way the “full” nonstandard hull can be coded in $\mathrm {BST}$ . Unfortunately, the coding involved is far from being a “morphism” in any sense, so the resulting opaque code is unsuitable for transferring nonstandard intuitions about the hull to its coded version. One point of working with subuniverses is that they have a natural coding (by standard sets).

Perhaps the most serious objection to this way of constructing nonstandard hulls is that $(\mathbf {B}_{\max }/\mathbf {E}_{\max }, \mathbf {D}_{\max }/\mathbf {E}_{\max })$ is just “too large.” Trivially, every nonstandard hull $(\mathbf {B}_{w}/\mathbf {E}_{w}, \mathbf {D}_{w}/\mathbf {E}_{w})$ of $(M, d)$ considered in Section 4.1 embeds isometrically into $(\mathbf {B}_{\max }/\mathbf {E}_{\max }, \mathbf {D}_{\max }/\mathbf {E}_{\max })$ (note that $\mathbf {B}_w = \mathbf {B}_{\max } \cap \mathbf {S}_w$ , $\mathbf {E}_w = \mathbf {E}_{\max }\cap \mathbf {S}_w$ , and $\mathbf {D}_w = \mathbf {D}_{\max } \cap \mathbf {S}_w$ ). In many interesting cases, the metric space $(M,d)$ has nonstandard hulls of arbitrarily large cardinality, so $\mathbf {B}_{\max }/\mathbf {E}_{\max }$ is not of standard size. It would be difficult to do further work with nonstandard hulls using this method, such as compare them with other standard metric spaces, take their products, or form the space of continuous functions on them. They are analogous to the “universal group” that can be constructed as a direct sum (or product) of all groups. This “object” is a proper class in $\mathrm {ZFC}$ , hardly if ever used for more than bookkeeping purposes. Another important point about working with subuniverses is that the objects produced are standard sets (or external sets of standard size).

6.2 Vakil’s construction

In [Reference Vakil30], Vakil presents a construction of nonstandard hulls of uniform spaces in $\mathrm {IST}$ . His method does not require fixing a particular subuniverse, and Standardization is used in a way similar to this paper. But Vakil’s method applies only to a certain class of uniform spaces, the so-called Henson–Moore spaces. Revealingly, these are precisely the spaces whose nonstandard hull is independent of the choice of the nonstandard universe, i.e., it is unique up to isomorphism and of standard size; see [Reference Henson and Moore8, Reference Vakil32].

6.3 Loeb measures in $\mathrm {IST}$

Diener and Stroyan [Reference Diener and Stroyan5, p. 274] outline a possible construction of Loeb measures in $\mathrm {IST}$ , referencing [Reference Stroyan and Bayod29, Section 2.2] for further details. Here we briefly consider this approach.

Let $(\Omega , \mathcal {A}, \mu )$ be an internal finitely additive measure space, with $\Omega \subseteq O$ for a standard set O. We take $\mathcal {A} = \mathcal {P}(\Omega )$ for simplicity, and analyze Loeb’s construction from the point of view of $\mathrm {BST}$ . The first step is to extend the algebra $\mathcal {A}$ to an external $\sigma $ -algebra. Bounded Idealization in $\mathrm {BST}$ implies that an externally countable union of (internal) sets is either equal to a finite union or is not internal. So the construction has to deal with external sets from the very beginning. The only way to treat external sets as objects in $\mathrm {BST}$ is via some kind of coding by sets. For example, every external sequence of (internal) sets has an extension $\langle X_n \,\mid \, n \in \mathbb {N} \rangle $ to an (internal) sequence,Footnote ⁷ which can be regarded as its code (of course an external sequence has many codes). This coding could be extended to higher levels of the Borel hierarchy over the algebra of (internal) subsets of $\Omega $ . A simpler solution, proposed in [Reference Diener and Stroyan5, Reference Stroyan and Bayod29], is to code Loeb measurable sets with the help of Souslin schemata. One can define a Souslin schema in $\mathrm {BST}$ as a function $S : \mathbb {N}^{<\omega } \to \mathcal {P} (\Omega ).$ Let $\mathcal {F} = \mathbb {N}^{\omega }$ . The kernel of S is the external set

$$ \begin{align*}\boldsymbol{\ker} S = \bigcup_{f \in \mathcal{F} \cap \mathbf{S}}\; \bigcap_{n \in \mathbb{N} \cap \mathbf{S}} S_{f \upharpoonright n}.\end{align*} $$

The external sets obtainable as kernels of Souslin schemata are Henson sets and Henson sets whose complement in $\Omega $ is also Henson are the Loeb sets. Loeb sets form the smallest external $\sigma $ -algebra $\boldsymbol {\sigma }(\mathcal {A})$ containing $\mathcal {P}(\Omega )$ ; see [Reference Stroyan and Bayod29, Theorem 2.2.3] (Luzin Separation Theorem). Let $\mathbf {L}$ be the external set of all pairs $\langle S_1, S_2\rangle $ such that $\boldsymbol {\ker } S_1 \cap \boldsymbol {\ker } S_2 = \emptyset $ and $\boldsymbol {\ker } S_1 \cup \boldsymbol {\ker } S_2 = \Omega $ . Then $\mathbf {L}$ can be viewed as a set of codes for Loeb sets. The algebraic operations (union, complement, and externally countable union) can be coded by external relations on $\mathbf {L}$ , and the Loeb measure itself can be coded by an external relation on $\mathbf {L} \times (\mathbb {R} \cap \mathbf {S})$ .

The objections raised in Section 6.1 against “full” nonstandard hulls apply also to the above approach to Loeb measures. In particular, one cannot form the quotient space of $\mathbf {L}$ modulo the relation in which two codes are equivalent when they code the same external set, and the coding of the set-theoretic operations is not a “morphism” (e.g., the code of the union of two sets is in no sense the union of their codes). It would be awkward to work with random variables on $\mathbf {L}$ via codes, and collections of random variables are even more challenging. In addition, it is customary in the literature (see [Reference Albeverio, Høegh-Krohn, Fenstad and Lindstrøm1, Reference Kanovei and Reeken14]) to regard not $\boldsymbol {\sigma }(\mathcal {A})$ but its completion $\boldsymbol {L}(\mathcal {A})$ as the Loeb $\sigma $ -algebra. It is not usually possible to extend $\mathbf {L}$ to an external set of codes for $\boldsymbol {L}(\mathcal {A})$ (not even in $\mathrm {HST}$ , because the Power Set axiom for external sets does not hold there). On these grounds, it is arguable whether the claim that this method represents the Loeb measure space is justified.

6.4 Bounded internal set theory

Following upon ideas of Lindstrøm [Reference Lindstrøm16], Diener and Stroyan [Reference Diener and Stroyan5] aim for “a description of the common ground between the two approaches to Robinson’s theory.” This takes the form of working in a polysaturatedFootnote ⁸ nonstandard universe $\mathfrak {N} = ( V(S), V({}^{\ast }\! S), \ast )$ axiomatically. They formulate Bounded Internal Set Theory (b $\mathrm {IST}$ ) and show that its axioms hold in such universes. This theory should not be confused with $\mathrm {BST}$ . The main differences are:

• • The language of b $\mathrm {IST}$ is closely tied to the structure $\mathfrak {N}$ . It contains a constant symbol for every internal set in $V({}^{\ast }\! S)$ and more; in particular, it is uncountable.

• • The axiom schemata T, I, and S are modified so as to apply only to those formulas in which all quantifiers are bounded.

• • The axioms of $\mathrm {ZFC}$ are not postulated (in fact, Replacement fails in $V(S)$ ).

b $\mathrm {IST}$ proves many results familiar from $\mathrm {IST}$ or $\mathrm {BST}$ , but, just like in these theories, its variables range over internal sets. It does not provide access to higher-order external sets without some coding. Of course, the nonstandard universe $\mathfrak {N}$ does provide external sets that can be used to construct nonstandard hulls and Loeb measures in the usual way, but then one is using the model-theoretic rather than the axiomatic approach. Overall, b $\mathrm {IST}$ is more a useful tool for work with superstructures than a self-standing axiomatics for nonstandard analysis.

6.5 Measure and integration over finite sets

Nelson’s Radically Elementary Probability Theory [Reference Nelson25] works with (hyper)finite probability spaces only. It requires only very elementary axioms (see [Reference Nelson25, Chapter 4]) that are easy consequences of the axioms of $\mathrm {IST}$ , or even of the weaker and more effective theory SCOT (see footnote 6). Further development of this approach can be found, e.g., in [Reference Cartier and Feneyrol-Perrin2, Reference Cartier and Perrin3], who study both Nelson’s S-integral on finite measure spaces (which they call Loeb–Nelson integral) and its refinement, which they call Lebesgue integral. The “radically elementary” approach is simple and elegant, and many interesting results have been obtained in this way.

It is beyond the scope of this paper to try to compare the “radically elementary” approach to the nonstandard measure theory based on the work of Loeb. Nelson shows [Reference Nelson25, Appendix 1] that for every stochastic process there is a nearby elementary process, and that theorems of the conventional theory of stochastic processes can be derived from their elementary analogues. The works [Reference Cartier and Feneyrol-Perrin2, Reference Cartier and Perrin3, Reference Nelson25] do not address the question whether (up to isomorphism) Loeb measure spaces can be obtained from their constructions.