TITLE: Why is the Monotone Convergence Theorem restricted to a nonnegative function sequence?
QUESTION [15 upvotes]: Monotone Convergence Theorem for general measure:
Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued $\Sigma-$measurable functions, i.e. for every $k\ge 1$ and every $x$ in $X$, $$0 \le f_k(x) \le f_{k+1}(x).$$ Next, set the pointwise limit of the sequence ${f_n}$ to be $f$. That is, for every $x$ in $X$, $$f(x) = \lim_{k\to \infty}f_k(x).$$ Then $f$ is $\Sigma-$measurable and $$\lim_{k\to \infty}\int f_k d\mu = \int f d\mu.$$

I've noticed that when it comes to monotone convergence theorem (either Lebesgue or general measure), usually its definition restricts the monotone function sequences to be nonnegative. I'm not sure why the 'nonnegative' is necessary.

REPLY [6 votes]: If they could be negative, then the statement would also have to be true for any sequence $f_n$ where $f_n(x)\geq f_{n+1}(x)$ by just reflecting across $0$. But then consider when $f_n:[0,\infty]\rightarrow R, f_n(x)=\frac{x}{n}$.  
In this case, for each $x$, $\lim_n f_n(x)=0$, and thus $f_n \rightarrow f=0$ pointwise, but the integrals are infinite for each finite $n$.