This answer should perhaps be posted as a comment, since the whole topic was already debated in this [MathOverflow Q&A](https://mathoverflow.net/questions/296312/do-you-know-important-theorems-that-remain-unknown/296839#296839). However i briefly restate the main result here and leave the above link for further detals: a definitive answer is Cafiero's convergence theorem (see [1]) which, roughly states that$\DeclareMathOperator{\Dm}{\operatorname{d\!}}$
$$
\int_X f_n\Dm\mu_n - \int_X f\Dm\mu_n \to 0 \iff \text{$(f_n\cdot\mu_n)_{n\geq 1}$ is uniformly exaustive.}
$$
Note that this *necessary and sufficient condition* is not very well known, even in the circles of measure theorists.

**Reference**

[1] Cafiero, F. (1953), "[Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi](http://www.numdam.org/item?id=RSMUP_1953__22__223_0)" [On the passage to the limit under the sign of integral for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 22: 223–245, [MR0057951](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0057951), [Zbl 0052.05003](https://zbmath.org/?q=an%3A0052.05003).