Proposition 7

*Let $f$ be a locally $\beta$-integrable function on $X$, $\nu$ a measure on $X$, and $\mu$ a measure on $G$. Assume that: (i) $\mu$ and $f$ are convolvable and formula (3) of No. 1 defines locally $\beta$-almost everywhere a convolution product $\mu *_{\beta} f$. (ii) $\chi \cdot \check{\mu}$ and $\nu$ are convolvable. (iii) The function $g(s, x) = f(s^{-1}x)\chi(s^{-1})$ is $(\mu \otimes \nu)$*-integrable. Then $f$ is essentially integrable for $(\chi \cdot \check{\mu}) * \nu$, the function $\mu *_{\beta} f$ defined by (3) is $\nu$*-integrable, and*

Since $g(s, x)$ is integrable for $\mu \otimes \nu$, the function $f(sx)$ is essentially integrable for $(\chi \cdot \check{\mu}) \otimes \nu$ and $f$ is essentially integrable for $(\chi \cdot \check{\mu}) * \nu$. By the Lebesgue–Fubini theorem, $\mu *_{\beta} f = \int g(s, x) d\mu(s)$ is $\nu$-integrable and