From pointwise convergence to $L^1$ and almost everywhere convergence.

Question

How would you show that if we have two sequences of random variables that converge pointwise $X_n \rightarrow X$ and $Y_n \rightarrow Y$, then $X_n + Y_n$ converges almost everywhere and in the $L^1$ sense to $X+Y$. I was trying to show the almost everywhere case. I started by saying that since we have that $X_n \rightarrow X$ and $Y_n \rightarrow Y$ $\forall w \in \Omega$, then we have that $\lim_{n \rightarrow \infty} |X_n + Y_n - X - Y| \leq \lim_{n \rightarrow \infty} |X_n -X| + |Y_n - Y| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$ for all $w \in \Omega$ and so in particular this holds for all $w \not\in \Sigma$ where $P(\Sigma) = 0$, showing the almost everywhere convergence. Is this a right approach or what am I doing wrong here? For the $L^1$ convergence, I really don't have any idea of how to approach it. Thanks for your help!

Answer 1

Your proof is fine for almost sure convergence; in fact this has nothing to do with probability theory. The second statement is false. A standard example is $X_n=nI_{(0,1/n)},Y_n=Y=X=0$ on $(0,1)$ with Lebesgue measure.