The following is taken from the book "Probability with Martingales" by David Williams
Suppose that $T$ is a stopping time such that for some $N$ in $\mathbb{N}$ and some $\epsilon>0$, we have, for every $n$ in $\mathbb{N}$ $$P(T\leq n+N|\mathcal{F}_n)>\epsilon,\hspace{12 mm}a.s.$$ Then $E(T)<\infty$.
My question is: why is $P(T\leq n+N|\mathcal{F}_n)$ a random variable (or what does he mean by "a.s" here)?