Expectation of submartingale always increasing?

Question

In a proof (upcrossing inequality) our professor used something, which i'm not sure i understand.

So let's say we have a $(\mathcal F_n)$-submartingale $(X_n)_{n\geq0}$ with $X_0 = 0$.

He then said that $E[X_n] \geq 0$ $\forall n \geq 1$, but how can you see this? I know that $E[X_{n+1} | \mathcal F_n] \geq X_n $, but how can i go from the conditional expectation to the "usual" expected value?

Answer 1

You can use the law of total expectation:

$$\mathbb{E}[X_{n+1}] = \mathbb{E}[\mathbb{E}[X_{n+1} \: | \: \mathcal{F}_n]] \geq \mathbb{E}[X_n]$$