Scholium

— For every function $f$ (resp. $g$) defined on $X$ (resp. $T$) with values in the Banach space $F$ or in $\overline{\mathbf{R}}$, let us denote by $\zeta(f)$ (resp. $\rho(g)$) the extension by 0 of $f$ to $T$ (resp. the restriction of $g$ to $X$). Then $\zeta(\rho(g)) = \varphi_X \cdot g$, $\rho(\zeta(f)) = f$. We denote by $\mu'$ the measure $\varphi_X \cdot \mu$ on $T$. For every $p \in [1, +\infty]$, Props. 1 and 2 imply that $\zeta$ maps $\mathscr{L}_F^p(X, \mu_X)$ into $\mathscr{L}_F^p(T, \mu')$, and that $\rho$ maps $\mathscr{L}_F^p(T, \mu')$ onto $\mathscr{L}_F^p(X, \mu_X)$, with preservation of norm in both cases, as well as of the integral when $p = 1$ (Th. 1); passing to the associated Hausdorff spaces, we obtain two isomorphisms inverse to each other. Similarly, if $\zeta$ and $\rho$ are applied to positive numerical functions, the essential upper integral is preserved (Prop. 1). Thus, if we agree to identify a function on $X$ with a function on T that is zero on X − T, and the measure $\mu_X$ with the measure $\mu'$, problems concerning induced measures are reduced to problems concerning measures defined by densities, treated in §5. This sort of reasoning is applicable to complex measures as well, by Cor. 3 of Th. 1.