# Strong law of large numbers --> convergence in L^{1}?! [duplicate]

Let $X_{1}, X_{2}, X_{3}, ...$ be a sequence of iid and integrable random variables with $E[X_{1}] = \mu$. Show that $$\frac{1}{n} (X_{1}+X_{2}+X_{3}+...+X_{n}) \to \mu$$ in $L^{1}$.

My thoughts so far: Strong law of large numbers implies almost surely convergence. So I still need to get from a.s. convergence to $L^{1}$ convergence. In my lectrure notes I have found two possible ways of doing so:

(a) If $\quad$ $X_{n} \to X \; a.s.$ $\quad$ and $\quad$ $E(|X_{n}|^{p}) \to E(|X|^{p})$ $\quad$ then $\quad$$X_{n} \to X in L^{p}. (b) If \quad X_{n} \to X \; in prob. \quad and \quad \{X_{n} : n \in \mathbb{N}\} is UI \quad then \quad$$X_{n} \to X$ in $L^{1}$.

Which is more promising? I tried both but...it didnt really work out...

• The iid hypothesis makes it trivial. For each $\epsilon>0$ there exists a $K$ such that the nice thing happens for all $X_n$. – kimchi lover Nov 11 '17 at 0:01