Remark

Under the hypotheses of the lemma, let us denote by abuse of language the function $h$ defined in the lemma by $f\circ g$.

If $g=g_1$ locally $\mu$-almost everywhere on X, the functions $g$ and $g_1$ have the same $\mu$-essential image and one has $f\circ g=f\circ g_1$.

If $g$ is a $\mu$-measurable function defined $\mu$-almost on X, one also denotes by $f\circ g$ the function $f\circ g_1$ where $g_1$ is a $\mu$-measurable function defined on X equal to $g$ locally $\mu$-almost everywhere.