Precisely, I am working on the monotone convergence theorem from the book of Folland, Real analysis. Statement is as follows:
If $\{f_n\}$ is a sequence in $L^+$ such that $f_j \leq f_{j+1}$ for all $j$ and $\displaystyle f = \lim_{n \to \infty} f_n$ $\displaystyle(=\sup_n f_n)$, then $\displaystyle\int f = \lim_{n \to \infty} \int f_n$.
Then, I did not understand what does it mean taking the limit of the sequence of functions. Also, I did not understand that how it equals to the supremum. Thanks.