Corollary 3

— Consider two positive numerical (resp. complex-valued) functions $f$ defined on $T$ and $f'$ defined on $T'$. If these functions are measurable for $\mu$ and $\mu'$, respectively, then the function

is measurable for $\mu \otimes \mu'$.

The case of complex functions, or of finite real functions, is an immediate consequence of Cor. 1. To treat the case of positive numerical functions, for every integer $n \geqslant 0$ we set $f_n = \inf(f, n)$, $f'_n = \inf(f', n)$, and we have (with the usual convention $0 \cdot (+\infty) = 0$) $f \otimes f' = \sup_n (f_n \otimes f'_n)$, whence the result.