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?
Best Answer
Depending on your course, showing that "the Lebesgue integral exists" could include showing that the integral (which can be defined for any non-negative measurable function) is finite. But it doesn't matter here: for any non-negative measurable (you didn't mention this; see also Juan's comment) function $g$, $\int g$ can be defined, and Fatou's lemma holds in that context, so that finiteness of the integral is established by afterwards by $$\int f\leq c.$$
In short: Your approach is correct.