Let $(\Omega,\mathscr{F},\mu)$ be a measure space and let $f:\Omega\to\mathbb{R}$ be a measurable function.
Suppose that the measure \begin{equation*} (f\mu)(A) = \int_A f(\omega)\mu(d\omega),\quad A\in\mathscr{F}, \end{equation*} is $\sigma$-finite. Is there a name for such functions $f$ that captures this property of $f\mu$ being $\sigma$-finite? I am not aware of one.
If $f\mu$ is $\sigma$-finite, then there are measurable sets $A_1,A_2,\ldots$ such that $\bigcup^\infty_{n=1}A_n = \Omega$ and for each $n$ \begin{equation*} (f\mu)(A_n) = \int_{A_n} f(\omega)\mu(d\omega) < \infty. \end{equation*} Intuitively this would suggest to me calling $f$ locally integrable, but this phrase is already reserved for something different. Another suggestion that comes to mind is $\sigma$-integrable, but judging from a quick view on wikipedia/Google scholar/the web this is not a thing.