I'm trying to work through the following problem.
Let $\{f_{n}\}_{n\geq 0}$ be a sequence of $L^{1}(\mu)$ functions (not necessarily positive) for some positive measure $\mu$ defined on a measurable space $(X,\mathfrak{M})$. Suppose that $f_{n}(x)\leq f_{n+1}(x)$ for all $x\in X$ and all $n\geq 0$, and that $\lim_{n\to\infty}f_{n}(x)=f(x)$, where $f\in L^{1}(\mu)$. Is it true that $\int_{X}f_{n}~d\mu\to\int_{X}f~d\mu$ as $n\to\infty$?
My thoughts: I'm not sure if my intuition is correct, but I do not think that the sequence of integrals $\int_{X}f_{n}~d\mu$ converges to $\int_{X}f~d\mu$. If I'm not mistaken, the given sequence should satisfy all but one hypothesis of the Monotone Convergence Theorem (namely that they are not assumed to be positive functions). I'm thinking that there's some counterexample, and was trying to play around with the definitions, but I'm pretty much stuck. Any help is appreciated!