15.1 Measurable Functions

Definition 15.1.1 (Measurable Function). Let $(X, \cm)$ and $(Y, \cn)$ be measurable spaces and $f: X \to Y$ be a mapping, then $f$ is $(\cm, \cn)$-measurable if $f^{-1}(E) \in \cm$ for all $E \in \cn$.

Definition 15.1.2 (Borel Measurable). Let $(X, \cm)$ be a measurable space, $Y$ be a topological space, and $f: X \to Y$ be a mapping, then $f$ is Borel measurable if $f$ is $(\cm, \cb_{Y})$-measurable.

Lemma 15.1.3. Let $X, Y$ be topological spaces and $f: X \to Y$ be continuous, then $f$ is Borel measurable.

Lemma 15.1.4. Let $(X, \cm)$ and $(Y, \cn = \sigma(\ce))$ be measurable spaces, and $f: X \to Y$ such that $f^{-1}(\ce) \subset \cm$, then $f$ is $(\cm, \cn)$-measurable.

Definition 15.1.5 (Generated $\sigma$-Algebra). Let $X$ be a set, $\bracs{(Y_i, \cn_i)}_{i \in I}$ be measurable spaces, and $\seqi{f}$ with $f_{i}: X \to Y_{i}$ for each $i \in I$. The $\sigma$-algebra generated by $\seqi{f}$,

is the smallest $\sigma$-algebra on $X$ such that each $\seqi{f}$ is measurable.