Remark

Let $\Lambda : t \mapsto \lambda_t$ be a $\mu$-adequate mapping of $T$ into $\mathscr{M}_+(X)$. If a mapping $\Lambda' : t \mapsto \lambda'_t$ of $T$ into $\mathscr{M}_+(X)$ is equal to $\Lambda$ locally $\mu$-almost everywhere, it follows at once from the definitions that $\Lambda'$ is also $\mu$-adequate, and that $\Lambda$ and $\Lambda'$ have the same integral. If now $H : t \mapsto \eta_t$ is a function with values in $\mathscr{M}_+(X)$, defined locally $\mu$-almost everywhere, we shall again say that $H$ is $\mu$-adequate if it is equal locally $\mu$-almost everywhere to a mapping $\Lambda : t \mapsto \lambda_t$ that is everywhere defined and $\mu$-adequate. We then set $\int \eta_t\, d\mu(t) = \int \lambda_t\, d\mu(t)$, a definition that does not depend on the function $\Lambda$ utilized. We leave to the reader the task of verifying that the propositions proved in the preceding Nos. extend to $\mu$-adequate functions defined locally $\mu$-almost everywhere.