Probability of a process being zero

Question

I found proofs in books (for example Oksendal's) where there is a process $S_t$ which satisfies $mathrm E[S_t^2]=0$ for all $t$. Based on this, a conclusion is made that
$$
mathbb P[S_t = 0 textrm{ for all } t in mathbb Q cap [0,T]]=1
$$
for some $T>0$ and where $mathbb Q$ is the set of rational numbers. Then, using the continuity of $S_t$ a further conclusion is made that
$$
mathbb P[S_t = 0 textrm{ for all } t in [0,T]] = 1
$$
My question is, why can one conclude the first probability but not the second from $mathbb E[S_t^2]=0$? Also, probably related, how do I show these conclusions rigorously?

Especially Lime · Accepted Answer

For a single $t$, we have $E(S_t^2)=0$ implies $P(S_tneq 0)=0$. Now if $t_1,t_2,ldots$ are any countable set of times, the probability that $S_{t_i}neq 0$ for some $i$ is at most the sum of the individual probabilities, i.e. $0$.
But we cannot in general make the same conclusion for uncountable sets of times; for example if $S_t$ is the indicator function of a Poisson point process then we have $E(S_t^2=0)$ for any given $t$, yet almost surely there are infinitely many values of $t$ with $S_t=1$.

Probability of a process being zero

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