The question I have received is the following:
Consider R equipped with a lebesgue measure. Let $f_1, f_2, ...$ be measurable functions on R with $f_1(x)\leq f_3(x)$ for all $x,f_2(x)\leq f_4(x)$ for all $x$, $f_3(x)\leq f_5(x)$ for all $x$, and so forth. Is it the case that $$lim_{j\to +\infty}\int f_j\:{d\mu(x)}=\int lim_{j\to +\infty}\: f_j\:{d\mu(x)}\:?$$
Either give a prove or a counterexample. What positive conclusion can you draw from these hypotheses?
So I attempted this question by providing the following counterexample:
I constructed the sequence of functions such that all the functions with even subscripts are the constant function 0, and all the odd functions are the constant function 1. Clearly, then the equality does not hold, so the first part of the question is complete, it is not the case. However, I am not sure what positive conclusion I can draw from these hypotheses, I am not quite sure what is expected? Am I looking at the question in the wrong way?
Non-negativity is missing in the hypothesis. Even if $\lim f_n$ exist equality may not hold. Example: $f_n =-I_{(n,\infty)}$ The entire sequence is increasing and $\lim f_n (x)=0$ for all $x$. But $\lim \int f_n =-\infty$ whereas $\int \lim f_n =0$.