Suppose that $\{X_k\}$ is a nonnegative submartingale, and $\Pr(X_1 = 0) = 0$. Then could we conclude that $\Pr(\liminf X_k=0) = 0$? What about $\Pr(\lim X_k=0) = 0$?
Some background information: This question arises when analysing the convergence of some randomized algorithm, where $X_k$ is the "distance" of the iterate from some "bad situation"; the desire is to prove that the iterate can avoid the bad situation with probability one.
No: let $\Omega=\{a,b\}$, $\mathcal{F}=2^\Omega$ and suppose that your filtration is $\mathcal{F}_1=\{\emptyset,\Omega\}$ and, for any $k\ge 2$, $\mathcal{F}_k=2^\Omega$. Let $\mathbb{P}$ be given by $\mathbb{P}(\{a\})=\mathbb{P}(\{b\})=\frac{1}{2}$.
Let $X_1\equiv 1$ and for $k\ge 2$ $X_k(a)=2$, $X_k(b)=0$. Then $(X_k)$ is a nonnegative converging martingale, with $\mathbb{P}(X_1=0)=0$, but as you can see $$\mathbb{P}(\lim\inf X_k=0)=\mathbb{P}(\lim X_k=0)=\frac{1}{2}$$