I am trying to prove Minkowski's inequality for integrals in the case $p = \infty$.
Suppose $(X_{1}, \mu_{1})$ and $(X_{2}, \mu_{2})$ are two $\sigma$-finite measure spaces and $f(x_{1}, x_{2})$ is a non-negative measurable function on $X_{1}\times X_{2}$ then $$\Big\|\int f(x_{1}, x_{2})\:\text{d}\mu_{2}\Big\|_{L^{\infty}(X_{1})}\leq \int \|f(x_{1}, x_{2})\|_{L^\infty(X_{1})}\:\text{d}\mu_{2}.$$
I am having trouble at the very first step in showing that $x_{2}\mapsto \|f(\cdot, x_{2})\|_{L^{\infty}(X_{1})}$ is measurable.
Some thoughts:
My idea is to write $\|f(\cdot, x_{2})\|_{L^{\infty}(X_{1})}$ as the limit of a sequence of measurable functions.
(i) The answer posted here suggests that it might be useful to use the fact that $\|g\|_{\infty} = \lim_{p\rightarrow\infty}\|g\|_{p}$ under certain conditions. However, the answer here suggests that in the case when $\|g\|_{\infty} < \infty$ then we need in addition that $\|g\|_{p_{0}} < \infty$ for some $p_{0}\in(0, \infty)$ (which may not be the case). Could we somehow make use of the $\sigma$-finiteness of the spaces?
(ii) The answer here suggests that if $h$ is a non-negative measurable function then $$\|h\|_{\infty} = \sup\Big\{\int hg\:\text{d}\mu : g\geq 0, \|g\|_{1} = 1\Big\}.$$ In this case, for $\mu_{2}$-a.e. $x_{2}$ we may write $$\|f(\cdot, x_{2})\|_{L^{\infty}(X_{1})} = \lim_{m\rightarrow\infty}\int f(x_{1}, x_{2})g_{m}(x_{1}, x_{2})\:\text{d}\mu_{1}$$ for some sequence of non-negative functions $\{g_{m}\}$ such that for each $m$ and $\mu_{2}$-a.e. $x_{2}$ we have $\|g(\cdot, x_{2})\|_{L^{1}(X_{1})}\leq 1$. However, I am not able to make this argument work since it is not clear that the functions $g_{m}$ are measurable with respect to the product measure on $X_{1}\times X_{2}$.
Any help is appreciated. Thank you.