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Unformatted text preview: continuous if for every open set V ⊆ Y, the set U = f −1 (V ) ⊆ X is open. A function f : X → Y is called measurable if f −1 (B ) ∈ B (X) for every B ∈ B (Y). Since the open sets generate the Borel sets, this theorem follows easily. Theorem 22.1. Suppose that (X, B (X)) and (Y, B (Y)) are topological measure spaces. The function f : X → Y is measurable if and only if f −1 (O) ∈ B (X) for every open set O ∈ B (Y). The next theorem tells us that continuous functions are necessarily measurable functions. Theorem 22.2. Suppose that (X, B (X)) and (Y, B (Y)) are topological measure spaces. If f : X → Y is continuous, then f is measurable. Proof. By definition, f : X → Y is continuous if and only if f −1 (O) ⊆ X is an open set for every open set O ⊆ Y. Since an open set is necessarily a Borel set, we conclude that f −1 (O) ∈ B (X) for every open set O ∈ B (Y). However, it now follows immediately from the previous theorem that f : X → Y is measurable. The following theorem shows that the composition of measurable functi...
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