Corollary 2

Let $\varphi$ be a central function on G (that is, such that $\varphi (gh) =$ $\varphi (hg)$ for all $g$ and $h$ in G) with values in a Banach space or in $\overline{\mathbf{R}}$.

a$)\varphi$ is measurable if and only if its restriction to T is measurable.

b$)\varphi$ is integrable if and only if the function $(\varphi |T)\delta_G$ is integrable on T, and in that case we have

Denote by $p: G/T\times T\rightarrow T$ the second projection. We have $\varphi \circ f$ = $(\varphi |T)\circ p$; further, the image under $p$ of the measure $\mu\otimes \frac{1}{w(G)}\delta_Gdt$ is $\frac{1}{w(G)}\delta_Gdt$. The corollary now follows from Th. 1 above and Th. 1 of Integration, Chap. V, §6, no. 2, applied to the two proper maps $f$ and $p$.