I tried to think about the question by this approach: what if there is a sequence of measures $\mu_n$ which uniformly converge to $\mu$ and we had all the hypothesis of Lebesgue's Dominated Convergence Theorem like:
Suppose that ${f_n}$ is a sequence of measurable functions where each $f_n$ is measurable regarding to $\mu_n$, that $f_n \to f$ point-wise almost everywhere as $n\to \infty$, and that $|f_n|\to g$ for all n, where $g$ is integrable.
then do we have the result of LCT here? i mean, is $f$ integrable and do we have the equation below? $$\int f d\mu=\lim_{n\to \infty} \int f_nd\mu_n$$