I have very little experience with probability and I would like some help in the following exercise: suppose $(X_n)$ is a sequence of identically distributed independent real-valued random variables, with $E|X_i|<\infty$. Because they are independently distributed, it follows that $EX_n = EX_1$ for all $n$ (I thought that this would be true because they are identically distributed, however I don't know how to prove it).
I must show that the sequence $\overline{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ converges in $L^1$ to $EX_1$. Can someone give me any hints or references on how to do this?
By the Weak Law of Large Numbers, you know that $\bar X_n$ converges to $EX_1$ in probability. If you manage to prove that the sequence $\bar X_n$ is uniformly integrable, you can thus conclude that it converges in $L^1$ as well.
To prove the uniform integrability of the sample mean $\bar X_n$, first prove that the sequence $X_n$ is uniformly integrable. Then, mainly using triangle inequality, you should be able to prove that $\bar X_n$ is also uniformly integrable and get your desired result.