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What the conditions, other than DCT and MCT, under which $$\lim_{n\to\infty} \int f_n(x) \ \mathsf dx = \int lim_{n\to\infty} f_n(x) \ \mathsf dx\quad $$
where the $f_n$ are measurable functions?
DCT- Dominated Convergence Theorem MCT- Monotone Convergence Theorem
If the integrals may be interpreted as definite integrals, then we have in fact iterated limits, because the integrals are limits themselves too. Therefore the following issues are relevant to the question:
