- I have a sequence $f_n$ of measurable, positive functions over some common domain $A$. I don't know whether they are increasing, yet I know that they converge to an integrable function $f$ on $A$ for which $f\ge f_n$ for every $f_n$ in my sequence.
Given: Theorem: If $(f_n)$ is increasing and each $f_n$ is positive on $A$ and $f_n \rightarrow f$ pointwise on $A$, except possibly on a subset of measure 0, then $\lim \int_Af_n=\int_A\lim f_n$.
My question, since I don't know whether the sequence in 1) is increasing, am I allowed to arrange the sets to that they are and then apply the theorem? I realize that this is probably more of a logic question than an analysis question.