Take the 2-minute tour ×  Hi, I have a question on my homework.
For each positive integer $n$, let $f_n:\mathbb{R}\to\mathbb{R}$
be integrable, $ ~f_n\geq 0$ and $f_n(x)\to f$ pointwise. I need
to show that if $\int f_n$ converges to some finite $c\geq 0$,
then $\int f$ exists and $0\leq\int f\leq c$.  I am thinking
that for an arbitrary function $f$ , the Lebesgue integral
exists iff $f$ is Lebesgue integrable or $\int f$ is infinite
(is this correct?). However, for a nonnegative function $f$, the
Lebesgue integral always exists and $\int f = \sup\{\int g：0\leq
g\leq f, ~~g$ bounded and supported on a set of finite
measure$\}$.  If what I am thinking is correct, then the
question seems quite straigtforward. We have $f_n(x)$ converges
to $f(x)$ for every $x$ and $f_n\geq 0$ for every $n$. So $f$ is
nonnegative everywhere and $\int f$ exists since the Lebesgue
integral exists for all nonnegative functions. That $0\leq\int
f\leq c$ just follows from the fact that $f \leq 0$ and Fatou's
Lemma.  Can someone tell