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. Consider this sequence.  Let $f_1$ be the indicator function on $[0,1/2]$, then $f_2$ be the indicator function on $[1/2,1]$. Now go back to zero and go by quarters to get $f_3$ through $f_6$. Continue in this fashion with 1/8, 1/16, etc. 
This sequence will converge to 0 in $\mathcal{L}^1$ but it will diverge by a.e. pointwise. This will occur at any number that is not a dyadic rational.
