In a former exam there was a multiple choice question about statements asking which one was true and the other 3 were wrong.
The following statement was said to be false but i can't understand why or find any counter-examples for which this statement is false. If anyone could help.
if $\{ f_n : R \rightarrow [-1,1] \} $ a sequence of measurable functions with $f_1 \leq f_2 \leq \cdots$ and $f_n$ converges pointwise to an absolutely integrable function $f$, then $lim_{n -> \infty} \int_{R} f_n(x)dx = \int_R f(x)dx$.
If they are not integrable you can have a simple counter example of
$$ f_n=-1+\mathbf{1}_{(-\infty,n)} \quad \text{and} \quad f(x)=0. $$
You can see that $\int_{\mathbb{R}}f=0$ and $\int_{\mathbb{R}}f_n= \int_{n}^\infty-1 dx=-\infty$. By the Beppo-Levi monotone convergence theorem, the counter example must be somehow negative. I also think that linearity of the integral and said theorem, must mean that the counter example comes from non-integrable functions.