My question is the following:
Consider a sequence $\{\,f_{n}\}$ of Lebesgue integrable functions over $\mathbb R$ that converges uniformly. Assume furthermore that $$ \int_{\mathbb R} f_{n}\,dx\,\to\,c\in\mathbb R. $$ It is not necessary that if the limit function $f$ is integrable its integral has the same value, i.e. $\lim \int f_{n} \neq \int \lim f_{n}$, see for example the function $f_{n} = \frac{1}{n} \chi_{[-n,n]}$ with $\int f_{n} = 1$ for all $n \in N$, but $\int \lim f_{n} = 0$. However, the function $f$ is itegrable. Is this true in general, is this function $f$ always integrable?
See remark under: I actually want to assume that all $f_{n}$ are positive (and so $f$ too).
Let $f_n=\sum_{k=1}^n \frac{(-1)^k}{k}\chi_{[k-1,k]}$.
Then $f_n$ converges uniformly, the $f_n$'s are integrable, $\int f_n$ also converges, but the limit does not belong to $L^1$.
If you assume that $f_n\ge 0$, then Fatou's Lemma provides that $$ \int_{\mathbb R}f=\int_{\mathbb R}\liminf_{n\to\infty}\, f_n\le \liminf_{n\to\infty}\int_{\mathbb R} f_n, $$ and hence $f\in L^1(\mathbb R)$.