non-negative supermartingale and the expectation of its terminal element

Question

Let $M_t$ be a non-negative supermartingale. Non-negative supermartingales converge, so we have $M_t \to M_\infty$.

Now, given $EM_\infty$ = $EM_0$, how can we conclude that $EM_t = EM_0$. It makes sense, but I don't know how to prove it. Any ideas?

Answer 1

$EM_t$ is a decreasing function of $t$. So $EM_{\infty} \leq EM_t \leq EM_0$. If $EM_{\infty} = EM_0$ then $EM_t=EM_0$ for all $t$.

Some details: $E(M_s |\mathcal F_t) \leq M_t$ if $t \leq s$. take expecattion on both sides to get $EM_s \leq EM_t$ for $t \leq s$. So $EM_T$ is a decreasing function. By Fatou's Lemm a $EM_{\infty} \leq \lim \inf EM_t =\inf_t EM_t$.