Property of submartingale and supermartingle?

Question

Is it true that for a submartingale, $$E(X_n) \le E(X_m)$$ for $n \le m$.

And for a supermartingale, $$E(X_n) \ge E(X_m)$$ for $n \le m$.

If it is true, then why?

I feel confused because the definition says $$E(X_n| \mathcal F_m) \ge E(X_m)$$ and $$E(X_n| \mathcal F_m) \le E(X_m)$$ for submartingale and supermartingale when $n \ge m$?

Answer 1

Take the expectation of both sides of the definition. They should simplify...