Family bounded in $\mathcal{L}^1$ has limit a.e. Let $(X, \mathcal{F} , \mu )$ be a measure space. 
Suppose $\lbrace X_n \rbrace$ is a family of functions in $\mathcal{L}^1$, bounded in $\mathcal{L}^1$ i.e. there exist $K \geq 0 $ such that $$||X_n||_1 \leq K  \ \ \forall n $$Is it true that $$\lim_{n \to \infty} X_n$$ exists a.e. ? Why ? 

No. Take in $\mathcal L^1[0,1]$:
$$
X_{2n}(x)=1\quad\text{nad}\quad X_{2n-1}(x)=0.
$$
Worse example, in $\mathcal L^1[0,1]$, 
where not even a subsequence converges pointwise is
$$
X_n(x)=\sin nx, \quad n\in\mathbb N.
$$
