I am actually working on martingales and I got stucked in the following example:
Let $X_1, \dots , X_n$ be independent integrable random variables with $\mathbb E(X_i) \equiv 0$, and consider the partial sums process $$S_n = X_1 + \dots + X_n$$ Suppose that $F_n = \sigma(X_1, \dots , X_n)$. The goals is to show that $(S_n)_n$ is martingale. The proof is given as follow: $$ \mathbb E(|S_n|) \leq \sum^n_{k=1} \mathbb E(|X_k|) < \infty$$ There I don't understand why it should be less than $\infty$. I expectation is monotone but how this interacts here with the assumption that $\mathbb E(X_i) \equiv 0$? Many thanks for some help.
If $EX_i$ is defined and has a finite value then $E|X_i|<\infty$. This is a basic fact from measure theory. Since $E|S_n|$ is at most equal to the finite sum $E|X_1|+E|X_2|+\cdots +E|X_n|$ it is finite.
[$EX=EX^{+}-EX^{-}$ is defined only when one of the two terms is finite. If $EX$ is a real number then both the terms must be finite so $E|X|=EX^{+}+EX^{-}<\infty$].