A Generalized Definition of Measurability

Let $(\Omega , \mathcal{F}, \mu )$ denote a measure space, let $\Omega^{\prime }$ denote a nonempty set with $\mathcal{F}^{\prime }$ a $\sigma$-field of subsets of $\Omega^{\prime }$, and let $f: A \rightarrow \Omega^{\prime }$ (with $A \in \mathcal{F}$ nonempty) be $\mathcal{F} / \mathcal{F}^{\prime }$-measurable, i.e. $f^{-1}(A^{\prime }) \in \mathcal{F}$ for every $A^{\prime } \in \mathcal{F}^{\prime }$. Let $\nu : \mathcal{F}^{\prime } \rightarrow \overline{\mathbb {R}}$ denote the measure induced by $f$, given for all $A^{\prime } \in \mathcal{F}^{\prime }$ by $\nu (A^{\prime }) = \mu (\left\{ \omega \in A: f(\omega ) \in A^{\prime }\right\} ) = \mu \left(f^{-1}(A^{\prime })\right)$ (also written $\nu = \mu \circ f^{-1}$).

Regarding the measure $\nu$:

Modify (MF14) and prove it in the more general setting of this section.

(MF14): Let $(\Omega , \mathcal{F}, \mu )$ denote a complete measure space. Pick nonempty $A$ in $\mathcal{F}$, and let $f$ be defined $\mu$-a.e. on $A$ and $\mathcal{F} / \mathfrak {B}^{*}$-measurable. (a) If $g$ is defined $\mu$-a.e. on $A$ and $f=g$ $\mu$-a.e. on $A$, then $g$ is $\mathcal{F} / \mathfrak {B}^{*}$-measurable. (b) If $g$ is defined everywhere on $A$ and $f=g$ $\mu$-a.e. on $A$, then $g$ is $\mathcal{F} / \mathfrak {B}^{*}$-measurable.