I have a question about changing limit and integral. I know of the monotone convergence theorem, so if my sequence is greater zero and increasing, I can change integral and limit. My question now is, what if my sequence is only bounded from below, by a possible negative number, and increasing. Can I change integral and limit is this case? I am pretty sure I can't. Does something change if I have a probability space? Thanks in advance!
2026-05-15 10:43:28.1778841808
Sequence of functions who are bounded from below, change of limit and integral
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$-I_{(n, \infty)}$ is a counter example for the general case. In the case of finite measure $f_n$ increasing and $\geq -a$ implies $f_n+a $ is non-negative and increasing so $\lim \int (f_n+a) \to \int (f+a)$ and we can subtract $a \mu (X)$ from both sides to get $\lim \int f_n \to \int f$