Abstract
The Borel subsets of a complete separable metric space have a number of interesting and useful characteristics. For example, two uncountable Borel subsets of complete separable metric spaces are necessarily Borel isomorphic, in the sense that there is a Borel measurable bijection from one to the other whose inverse is also Borel measuable. A related result says that if we have a Borel measurable injection from one complete separable metric space to another, then the image under this map of each Borel subset of the domain is Borel. If the function involved here is Borel but not necessarily injective, then the images of Borel sets are measurable for every finite Borel measure on the range space.
This Chapter is devoted to proving such results and to showing the context in which they arise.
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Notes
- 1.
Let {Y α } be an indexed collection of sets such that
-
(a)
for each α the set Y α has the same cardinality as the set X α , and
-
(b)
\(Y _{\alpha _{1}}\) and \(Y _{\alpha _{2}}\) are disjoint if α 1≠α 2
(for instance, one might let Y α be X α ×{α}). The disjoint union of the X α ’s is defined to be the union of the Y α ’s. (One generally thinks of the Y α ’s as being identified with the corresponding X α ’s.)
-
(a)
- 2.
Suppose that X and Y are topological spaces. A function f : X → Y is open if for each open subset U of X the set f(U) is an open subset of Y.
- 3.
Recall that each ordinal α can be written in a unique way in the form \(\alpha =\beta +n\), where β is either zero or a limit ordinal and where n is finite. The ordinal α is called even if n is even and odd if n is odd.
- 4.
We will see (Theorem 8.3.7 ) that the injectivity and measurability of f imply that \(B \in \mathcal{B}(Y )\).
- 5.
Of course \(\mathcal{A}_{X_{0}}\) and \(\mathcal{B}_{Y _{0}}\) are the traces of \(\mathcal{A}\) and \(\mathcal{B}\) on X 0 and Y 0 (see Exercise 1.5.11).
- 6.
See Exercise 5 for a proof of Theorem 8.3.6 that does not depend on Proposition 8.2.13 or 8.3.5.
- 7.
A subset of a Hausdorff space is relatively compact if its closure is compact.
- 8.
This example assumes more Banach space theory than is developed in this book.
- 9.
Suppose that X is an infinite-dimensional Banach space. If the weak topology on X is metrizable, then there is an infinite sequence {f i } in X ∗ such that each f in X ∗ is a linear combination of f 1, …, f n for some n (choose {f i } so that for each weakly open neighborhood U of 0 there is a positive integer n and a positive number ε such that x ∈ U holds whenever x satisfies | f i (x) | < ε for i = 1, …, n; then use Lemma V.3.10 in [42]). Thus X ∗ is the union of a sequence of finite-dimensional subspaces of X ∗ . Since this is impossible (use Exercise 3.5.6, Corollary IV.3.2 in [42], and the Baire category theorem), we have a contradiction, and the weak topology on X is not metrizable. A similar argument shows that the weak ∗ topology on X ∗ is not metrizable.
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Cohn, D.L. (2013). Polish Spaces and Analytic Sets. In: Measure Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6956-8_8
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