Rasiowa–Sikorski lemma

The previous chapter was devoted to constructing objects by transfinite induction. A typical scheme for such constructions was a diagonalization argument like the following. To find a subset S of a set X concerning a family P = {Pα : α < κ} we chose S = {xξ ∈ X : ξ < κ} by picking each xξ to take care of a set Pξ. But what can we do if the cardinal κ is too big compared to the freedom of choice of the xξs; for example, if the set X has cardinality less than κ?

There is no absolute answer to this question. In some cases you can do nothing. For example, if you try to construct a subset S of ω different from every set from the family P(ω) = {Bξ: ξ < c}, then you are obviously condemned to failure. The inductive construction does not work, since you would have to take care of continuum many conditions, having the freedom to choose only count ably many points for S.

In some other cases you can reduce a family P to the appropriate size. This was done, for example, in Theorem 6.3.7 (on the existence of Bernstein sets) in which we constructed a nonmeasurable subset B of ℝn: The natural family P - ℒ of cardinality 2c was replaced by the family P0 = {Pξ: ξ < c} of all perfect subsets of ℝn.

We first review some notions and facts about continuous representations of

in a locally complete topological vector space V. In fact, all we need is the representation by right translations of G in C∞(G), L2(G), C∞(H\G), or L2(H\G) (H closed unimodular subgroup, mainly F). Much of this has been encountered earlier, implicitly or explicitly. All this is valid in a much more general framework (see e.g. [7]).

A continuous representation (π, V) of G into V is a homomorphism of G into the group of automorphisms of V that is continuous; in other words, the map (g, v) ↦ π(g).v is a continuous map of G × V into V. If V is a Hilbert space then it is said to be i>unitary if π (g) (g ∈ G) leaves the scalar product of V invariant. Then the operator norm ∥π(g)∥ is uniformly bounded (by 1), and it is known that continuity already follows from separate continuity:

(1) for every v ∈ V, the map g ↦ π(g).v of G into V is continuous (see e.g. [7, §3] or [10, §VIII.1]).

The assumption “locally complete” is made to ensure that if α ∈ Cc(G) then ∫G α(x)π(x).v dx converges to an element of V. More generally, ∫ µ(x)π(x).v converges if µ is a compactly supported measure on G (see an important example in 14.4).

This chapter is designed to help the reader to master the technique of recursive definitions. Thus, most of the examples presented will involve constructions by transfinite induction.

Measurable and nonmeasurable functions

Let B be a σ-algebra on ℝn. A function f: ℝ → ℝ is said to be a B-measurable function if f-1(U) ∈ B for every open set U ⊂ ℝ. Notice that if f is B-measurable then f-1(B) ∈ B for every Borel set B ⊂ ℝ. This is the case since the family {B ⊂ ℝ: f-1(B) ∈ B} is a σ-algebra containing all open sets.

We will use this notion mainly for the σ-algebras of Borel, Lebesguemeasurable, and Baire subsets of ℝn, respectively. In each of these cases B-measurable functions will be termed, respectively, as Borel functions (or Borel-measurable functions), measurable functions (or Lebesgue-measurable functions), and Baire functions (or Baire-measurable functions). Clearly, every continuous function is Borel-measurable and every Borel-measurable function is measurable and Baire.

A function f: ℝn → ℝ is non-Borel (or non-Borel-measurable) if it is not Borel. Similarly, we define non-Baire (-measurable) functions and non-(Lebesgue-)measurable functions.

Also recall that the characteristic function χA of a subset A of a set X is defined by putting χA(x) = 1 if x ∈ A and χA(x) = 0 for x ∈ X \ A.

The first theorem is a corollary to Theorems 6.3.7 and 6.3.8.

The course presented in this text concentrates on the typical methods of modern set theory: transfinite induction, Zorn's lemma, the continuum hypothesis, Martin's axiom, the diamond principle ◊, and elements of forcing. The choice of the topics and the way in which they are presented is subordinate to one purpose – to get the tools that are most useful in applications, especially in abstract geometry, analysis, topology, and algebra. In particular, most of the methods presented in this course are accompanied by many applications in abstract geometry, real analysis, and, in a few cases, topology and algebra. Thus the text is dedicated to all readers that would like to apply set-theoretic methods outside set theory.

The course is presented as a textbook that is appropriate for either a lower-level graduate course or an advanced undergraduate course. However, the potential readership should also include mathematicians whose expertise lies outside set theory but who would like to learn more about modern set-theoretic techniques that might be applicable in their field.

The reader of this text is assumed to have a good understanding of abstract proving techniques, and of the basic geometric and topological structure of the n-dimensional Euclidean space ℝn. In particular, a comfort in dealing with the continuous functions from ℝn into ℝ is assumed. A basic set-theoretic knowledge is also required.

Our main prerequisites may be listed as follows.

1.1Elementary theory of Lie groups and Lie algebras, and the interpretation of the elements of the Lie algebra as differential operators on the group or its coset spaces. This will be used only for SL2(ℝ), its Lie subgroups, and the upper halfplane. The book by Warner [58] is more than sufficient for our needs, except for some facts on Haar measures (2.9, 10.9).

1.2The regularity theorem for elliptic operators (see the remark in 2.13 for references).

1.3Some functional analysis, mainly about operators on Hilbert spaces. For the sake of definiteness, I have used two basic textbooks ([46] and [51]) and have given at least one precise reference for every theorem used. But this material is standard and the reader is likely to find what is needed in his (or her) favorite book on functional analysis. The demands will increase as we go along, and the material will be briefly reviewed before it is needed. Such review is mainly intended to refresh memory, fix notation, and give references, not to be a full-fledged introduction.

1.4Infinite dimensional representations of G. We shall review what we need in Sections 14 and 15 and also give a few proofs, but mostly refer to the literature. An essentially self-contained discussion of some of the results on SL2(ℝ) stated in Sections 2 and 14 is contained in the first part of [40].