# exponential stochastic process

Let $T$ be a random exponentially distributed time. $P \left(T > t \right)=e^{-t}$. Define $M$ via $M_t = 1$ if $t-T \in Q^+$, $M_t = 0$ otherwise. Where $Q^+$ being positive rationals. let $\{F_t\}$ be a filtration generated by the process $M$. We have to prove that for $t\ge s$ and and $F \in F_s$

$$E \left[ 1_{\{t-T \in Q^+\}} \right] = 0 = E \left[ M_S 1_F \right].$$

The LHS is $E(M_t)=P(T\in t+Q^+)=0$ because the Lebesgue measure of $t+Q^+$ is zero and the distribution of $T$ is absolutely continuous. The RHS is nonnegative and at most $E(M_s)=0$, hence zero. The conditions that $t\ge s$ and that $F\in F_s$ are irrelevant.