Category Measure Spaces

Abstract

It was shown in the last chapter that no one-to-one mapping of the line onto itself can map the nullsets onto the sets of first category and at the same time map the measurable sets onto the sets having the property of Baire. More generally, if (X, S, μ) is a measure space with 0 〈 μ(X) 〈 ∞, and Y is a separable metric space without isolated points, then no S-measurable mapping of X into Y can be such that the inverse image of every set of first category has measure zero. For if f were such a mapping we could define a finite nonatomic Borel measure ν in Y by setting v(E) = μ(f ^-1 (E)) for every Borel set E. By Theorem 16.5, Y could be decomposed into a ν-nullset and a set of first category. Then v(Y) would be 0, contrary to μ(X) 〉 0.