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)?


1 Answer 1


Conditional probabilities ARE random variables!

If you are dealing with martingales, then you have certainly seen conditional expectations before: $\mathbb{E}[X\mid\mathcal{F}]$ is a $\mathcal{F}$-measurable random variable $Y:\Omega\rightarrow\mathbb{R}$ such that for all $F\in\mathcal{F}$, $$ \mathbb{E}[Y;\ F]=\mathbb{E}[X;\ F], $$ where $\mathbb{E}[Z;\ F]:=\mathbb{E}[Z\cdot 1_{F}]$.

Conditional probabilities of the type that you describe are simply the conditional expectations of indicators: $$ P(T\leq n+N\mid \mathcal{F}_n):=\mathbb{E}[1_{T\leq n+N}\mid\mathcal{F}_n], $$ which is again a $\mathcal{F}_n$-measurable random variable on your original probability space.


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