Remark

— Let $(\pi, g)$ be a $\mu$-adapted pair, $\pi'$ a mapping of T into X, and $g'$ a finite numerical function $\geq 0$ defined on T, such that $\pi'$ (resp. $g'$) is equal to $\pi$ (resp. $g$) locally almost everywhere for $\mu$. Then the pair $(\pi', g')$ is $\mu$-adapted, the measures $\lambda_t = g(t)\varepsilon_{\pi(t)}$ and $\lambda'_t = g'(t)\varepsilon_{\pi'(t)}$ are equal locally almost everywhere, and $\int g(t)\varepsilon_{\pi(t)}\,d\mu(t) = \int g'(t)\varepsilon_{\pi'(t)}\,d\mu(t)$. If now $\pi'$ and $g'$ are only defined locally almost everywhere (for $\mu$) and if there exists a $\mu$-adapted pair $(\pi, g)$ such that $\pi'$ (resp. $g'$) is equal to $\pi$ (resp. $g$) locally almost everywhere, one again says that the pair $(\pi', g')$ is $\mu$-adapted and one sets

(cf. §3, No. 3, Remark). The statements of Ths. 1 and 2 and of Prop. 3 remain valid when $\pi$ and $g$ are only assumed to be defined locally almost everywhere.