Almost Everywhere

In (MF15), the completeness of ( $\Omega , \mathcal{F}, \mu$ ) is not a redundant assumption.

(MF15): Let $(\Omega , \mathcal{F}, \mu )$ denote a complete measure space, and let $A \in \mathcal{F} - \left\{ \varnothing \right\}$. Let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ denote a sequence of $\mathcal{F} / \mathfrak {B}^{*}$-measurable functions, each of which is defined $\mu$-a.e. on $A$. Suppose that $f$ is defined $\mu$-a.e. on $A$ and that $\lim_{n} f_{n}(\omega ) = f(\omega )$ $\mu$-a.e. on $A$. Then $f$ is $\mathcal{F} / \mathfrak {B}^{*}$-measurable.

Suppose that $f: \mathbb {R} \rightarrow \mathbb {R}$ is a differentiable function. Then $f^{\prime }$ is a Borel measurable function.

Let $f: \mathbb {R} \rightarrow \overline{\mathbb {R}}$. The statement $f$ is continuous at almost every $x \in \mathbb {R}$ neither implies nor is implied by the statement there exists continuous $g: \mathbb {R} \rightarrow \overline{\mathbb {R}}$ such that $f=g$ $\mu$-a.e. on $\mathbb {R}$.