# Schröder–Bernstein theorem for measurable spaces

Contents 1 The theorem 1.1 Comments 1.2 Proof 2 Examples 2.1 Example 1 2.2 Example 2 3 References The theorem Let {displaystyle X} and {displaystyle Y} be measurable spaces. If there exist injective, bimeasurable maps {displaystyle f:Xto Y,} {displaystyle g:Yto X,} then {displaystyle X} and {displaystyle Y} are isomorphic (the Schröder–Bernstein property).

Comments The phrase " {displaystyle f} is bimeasurable" means that, first, {displaystyle f} is measurable (that is, the preimage {displaystyle f^{-1}(B)} is measurable for every measurable {displaystyle Bsubset Y} ), and second, the image {displaystyle f(A)} is measurable for every measurable {displaystyle Asubset X} . (Thus, {displaystyle f(X)} must be a measurable subset of {displaystyle Y,} not necessarily the whole {displaystyle Y.} ) An isomorphism (between two measurable spaces) is, by definition, a bimeasurable bijection. If it exists, these measurable spaces are called isomorphic.

Proof First, one constructs a bijection {displaystyle h:Xto Y} out of {displaystyle f} and {displaystyle g} exactly as in the proof of the Cantor–Bernstein–Schroeder theorem. Second, {displaystyle h} is measurable, since it coincides with {displaystyle f} on a measurable set and with {displaystyle g^{-1}} on its complement. Similarly, {displaystyle h^{-1}} is measurable.

Examples Example maps f:(0,1)→[0,1] and g:[0,1]→(0,1). Example 1 The open interval (0, 1) and the closed interval [0, 1] are evidently non-isomorphic as topological spaces (that is, not homeomorphic). However, they are isomorphic as measurable spaces. Indeed, the closed interval is evidently isomorphic to a shorter closed subinterval of the open interval. Also the open interval is evidently isomorphic to a part of the closed interval (just itself, for instance).

Example 2 The real line {displaystyle mathbb {R} } and the plane {displaystyle mathbb {R} ^{2}} are isomorphic as measurable spaces. It is immediate to embed {displaystyle mathbb {R} } into {displaystyle mathbb {R} ^{2}.} The converse, embedding of {displaystyle mathbb {R} ^{2}.} into {displaystyle mathbb {R} } (as measurable spaces, of course, not as topological spaces) can be made by a well-known trick with interspersed digits; for example, g(π,100e) = g(3.14159 265…, 271.82818 28…) = 20731.184218 51982 2685….

The map {displaystyle g:mathbb {R} ^{2}to mathbb {R} } is clearly injective. It is easy to check that it is bimeasurable. (However, it is not bijective; for example, the number {displaystyle 1/11=0.090909dots } is not of the form {displaystyle g(x,y)} ).

References S.M. Srivastava, A Course on Borel Sets, Springer, 1998. See Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94). Categories: Theorems in measure theoryDescriptive set theoryTheorems in the foundations of mathematics

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