$f_n \rightarrow f$ pointwise a.e. does not hold for integral

Question

Let $\lambda$ be the Lebesgue measure on $(\mathbb R, \mathcal M_{\lambda^*})$ and $f, f_n: \mathbb R \rightarrow \mathbb R$.

I know that the following statement is not true in general:

If $f_n, f$ are integrable and $f_n(x) \rightarrow f(x)$ pointwise a.e., then: $\int f_n d \lambda \rightarrow \int f d \lambda$.

But is there any example which really shows that the statement is not true?

Answer 1

Try $f_n(x) = n\chi_{(0,\frac{1}{n})}(x)$ with the usual Lebesgue measure.