I am looking for a reference from a book for the result of continuity of an integral (found in https://www.encyclopediaofmath.org/index.php/Parameter-dependent_integral):
Let $D \subset \mathbb{R}^n$ and $t \in (0,T)$, and let $f(x,t)$ be a function that is continuous in $t$ for almost every $x \in D$, and let $|f(x,y)| \leq g(x)$ where $g$ is integrable. Then
$$J(t) = \int_{D}f(x,t)\,dx$$
is continuous with respect to $t$.
I tried the references on the site but they require continuity of $t \mapsto f(x,t)$ for every $x$, but I need it for almost every $x$ only.
Best Answer
It's rather straightforward to reduce it to the assumption that $t\mapsto f(x,t)$ is continuous for all $x$, one just needs to replace $f$ with $h(x,t) = f(x,t)\cdot \chi_{D\setminus N}(x)$, where $N$ is a null set such that $t\mapsto f(x,t)$ is continuous for all $x\in D\setminus N$. $J(t)$ isn't changed at all.
However, you ask for a citable reference, so:
gives a slightly more general formulation. (I don't know if the book has been translated, it should have been, it's a good book.)
The translation is mine: