Borel Isomorphism

Two measurable spaces (X, ) and (Y, ) are called isomorphic iff there is a one-to-one function f from X onto Y such that f and f−1 are measurable. Two metric spaces (X, d) and (Y, e) will be called Borel-isomorphic, written X ∼ Y, iff they are isomorphic with their σ-algebras of Borel sets.

Clearly, Borel isomorphism comes somewhere between being homeomorphic topologically and being isomorphic as sets, which means having the same cardinality. The following main fact of this section shows that in many cases, surprisingly, Borel isomorphism is just equivalent to having the same cardinality:

Theorem If X and Y are two separable metric spaces which are Borel subsets of their completions, then X ∼ Y if and only if X and Y have the same cardinality, which moreover is either finite, countable, or c (the cardinal of the continuum, that is, of [0, 1]).

Remarks In general, the continuum hypothesis, stating that no sets have cardinality uncountable but strictly less than c, is independent of the other axioms of set theory, including the axiom of choice (see the notes to Appendix A.3). For Borel sets in complete separable metric spaces, however, the continuum hypothesis follows from the axioms, by the theorem about to be proved. Examples of the isomorphism are ℝ ∼ ℝ2 and ℝ ∼ ℝ∖, the space of irrational numbers.