In https://en.wikipedia.org/wiki/Monotone_convergence_theorem, limit needs to exists to make further conclusion. However, I think this is not necessary? One reason is for monotone function they always converge. So we can say limit of integral of sequence of function converge to integral of limit, which is unknown?
2026-04-13 19:54:36.1776110076
Weaker version of Monotone Convergence Theorem when limit is not known?
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You are right, it is not necessary to assume the pointwise convergence of $(f_n)$. This is an implication of monotonicity. However, in order to state the theorem, the pointwise limit appears: $$ \lim_{n\to\infty} \int f_n \ d\mu= \int (\lim_{n\to\infty} f(x)) d \mu. $$ The theorem not only gives convergence of the integrals, it also specifies the limit.