Is it random variable?

Question

Let $(X_n, F_n)$ be martingale. Let $\tau$ be stopping time with respect to $(X_n,F_n)$.

Consider $X_{\tau (w)}(w)$. Is it random variable?

$X_{\tau (w)} = \sum_j X_j(w) 1_{[j,j+1)}(\tau(w))$

It is well defined. But I wonder if right hand side is measurable.

Could you help me?

Answer 1

Yes, it is measurable. Note that

$$\omega \mapsto X_j(\omega) 1_{[j,j+1)}(\tau(\omega)) = X_j(\omega) 1_{\{j \leq \tau < j+1\}}(\omega)$$

is measurable (as $X_j$ and $\tau$ are measurable) and therefore $$X_{\tau} = \lim_{k \to \infty} \sum_{j=0}^k X_j 1_{[j,j+1)}(\tau)$$ is measurable as pointwise limit of measurable functions.