21.1 Measurable Functions

Definition 21.1.1 (Measurable Function).label 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 21.1.2 (Borel Measurable).label 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.

Definition 21.1.3 (Convergence Almost Everywhere).label Let $(X, \cm, \mu)$ be a measure space, $Y$ be a topological space, $\seq{f_n}$ and $f$ be Borel measurable functions from $X$ to $Y$, then $f_{n} \to f$ almost everywhere/a.e. if there exists a $\mu$-null set $N \in \cm$ such that $f_{n} \to f$ pointwise on $X \setminus N$.

Lemma 21.1.4.label Let $X, Y$ be topological spaces and $f: X \to Y$ be continuous, then $f$ is Borel measurable.

Lemma 21.1.5.label 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 21.1.6 (Generated $\sigma$-Algebra).label 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.

Theorem 21.1.7 (Egoroff).label Let $(X, \cm, \mu)$ be a finite measure space, $(Y, d)$ be a metric space, and $\seq{f_n}$ and $f$ be Borel measurable functions from $X$ to $Y$ such that $f_{n} \to f$ almost everywhere, then for any $\eps > 0$, there exists $E \in \cm$ such that:

1. (1)

   $f_{n} \to f$ uniformly on $E$.

2. (2)

   $\mu(X \setminus E) < \eps$