Let $\mu$ be a regular $\sigma$-finite Borel measure on $X$. Suppose we have a sequence of continuous functions $$f_n: X \times [0,T] \rightarrow \mathbb{R}: (x,t) \mapsto f(x,t), $$ which converges pointwise to a continuous function $f: X \times [0,T] \rightarrow \mathbb{R}$. Suppose the following:
- There exists a $g: X \times [0,T] \rightarrow \mathbb{R}$ such that for each $t \in [0,T]$ the integral of $g(\cdot, t)$ exists and for all $n$: $f_n \leq g$,
- For each $t \in [0,T]$ we have that $$ \lim_{n \rightarrow \infty} \int_X f_n(\cdot,t) d\mu = \int_X f(\cdot,t), $$
- For each $x \in X$ the convergence $f_n(x,\cdot) \rightarrow f(x,\cdot)$ is uniform.
Is this enough to conclude that the convergence
$$ \lim_{n \rightarrow \infty} \int_X f_n(\cdot,t) d\mu = \int_X f(\cdot,t) $$
is uniform in $t$? Maybe a useful theorem to prove it would be Egonov's Theorem.
I found the article
Emanual Parzen, Some conditions for uniform convergence of integrals
where in the beginning of the article the author claims it is true and refers to a book. I looked into the book, but the notations and assumptions are too vague to me and the Theorem isn't stated in its proper form. I ask this question since I'm not sure if I interpreted the previous references correctly.