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在chap5.pdf中, 假定了完备测度空间
Theorem 3.1. Let $(X, S, μ)$ be a complete measure space
(1) For two functions $f$ and $g$ from $X$ to $\mathbb R$, if $f$ is measurable and $g=f$ almost everywhere, then $g$ is measurable.
Proof. (1) Let $E:=\{x \in X: f(x)=g(x)\}$. Then $E^c$ is a null set. For $a \in \mathbb{R}$, we have
\[
\{x \in X: g(x)>a\}=\{x \in E: g(x)>a\} \cup\left\{x \in E^c: g(x)>a\right\} .
\]
Moreover,
\[
\{x \in E: g(x)>a\}=\{x \in E: f(x)>a\}=\{x \in X: f(x)>a\} \backslash\left\{x \in E^c: f(x)>a\right\} .
\]
Since $\mu$ is a complete measure and $E^c$ is a null set, both sets $\left\{x \in E^c: g(x)>a\right\}$ and $\left\{x \in E^c: f(x)>a\right\}$ are measurable. Furthermore, the set $\{x \in X: f(x)>a\}$ is measurable, because $f$ is measurable. This shows that $\{x \in X: g(x)>a\}$ is measurable for every $a \in \mathbb{R}$. Therefore, $g$ is a measurable function. |
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