**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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