Proposition 2

Every numerical function $f$ essentially integrable for $m$ is essentially integrable for $u \circ m$, and $\int f d(u \circ m) = u(\int f dm)$.

The proposition is obvious, in view of the equality $y' \circ u \circ m = {}^t u(y') \circ m$ for all $y \in G'$.

In general, if $f \in \mathscr{L}(m)$, the integral $\int f dm$ belongs to ${F'}^*$ but not to $F$ (see the above Example). However: