CONSTRUCTING NONSTANDARD HULLS AND LOEB MEASURES IN INTERNAL SET THEORIES

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.