Proposition 4

For a mapping $f$ of $T$ into a topological space $G$ to be measurable for the measure $u \cdot \theta$, it is necessary and sufficient that the restriction of $f$, to the $\theta$-measurable set $S$ of the $t$ such that $u(t) \neq 0$, be $\theta$-measurable.

When $u$ and $\theta$ are positive, this follows at once from Prop. 3 of §4, No. 3. The result then extends to the case that $u$ and $\theta$ are complex thanks to Prop. 2.