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 me under what condition the Lebesgue integral exists? Is my attempt correct?