Condition satisfied on a set is conditional probability

Question

If I am told that $X(T)=a$ on $\{T\lt \infty\}$, a.s. Is that the same as the conditional probability $$P(X(T)=a|T\lt \infty)=1.$$

I know this is very basic but for some reason I haven't been able to find a concrete answer to this question.

Answer 1

It's not the same, but you do have an implication.

In particular, $$X(T)=a \text{ on } \{T < \infty\} \implies P(X(T)=a|T<\infty) = 1,$$ which, if we think purely in terms of the conditional probability, is just an example of the implication "If an event $A$ is sure to occur, then $P(A) = 1$." The opposite direction is not necessarily true.

That said, what you have left out of your question is that $T$ is almost certainly a hitting time for $X(t) = a,$ in which case we do have $X(T)=a$ on $\{T < \infty\}.$ This would make the converse vacuously true for this particular assignment of $T.$