Limit of an $L^1$ sequence

Question

We have functions $f_n\in L^1$ such that $f_n(x)$ tends to some $f(x)$ for almost all $x$. Does this mean that $f_n\to f$ in $L^1$? A necessary condition is $\|f_n\|<M$, is it sufficient?

Answer 1

Hint:

Consider $L^1(0,1)$, look at $f_n=n\cdot\chi_{(0,1/n)}$ (here $\chi_A$ denotes the characteristic function on $A$).

1. $f_n\to f$ where $f=...$.

2. $\|f_n\|\le ...$ and $\|f\|=...$

Answer 2

No. For example, if $f_n=\chi_{[n,n+1]}$ then $(f_n)_{n\ge1}$ has pointwise limit $0$ but $\|f_n\|_1=1$ for every $n$ so $f_n\not\to 0$ in $L^1(\mathbb{R})$.