Nonnegative local martingale

Every nonnegative local martingale is a supermartingale. In particular, one that starts from zero is identically zero: its nonnegative value at every time has expectation at most zero.

For $M_0=1$ and $M_t\to 0$, stop at the infimum of strict crossings above $a>1$. Continuity bounds the stopped process by $a$ and makes its value at a finite crossing time exactly $a$. Its expectation remains one, and bounded convergence at infinity yields $\mathbb{P}(\sup M>a)=1/a$. The maximum has density $m^{-2}$ on $m>1$, with no endpoint atom.

A nonnegative local martingale is a supermartingale. If its integrable terminal limit has the same expectation as its initial value, conditional Fatou lemma gives $\mathbb{E}[Z_{\infty }\mid {\mathcal{F}}_t]\le Z_t$, and equality of expectations makes this an equality. It is therefore a uniformly integrable martingale.