I am hopelessly stuck on the following problem.
Consider a filtrated propability space $(\Omega, \mathcal{A}, \mathbb{F} = (\mathcal{F}_t)_{t \geq 0}, P)$. Let $X$ be an adaptated càd stochastic process. Let further $T$ be a stopping time and $$\mathcal{F}_{T \wedge s} = \{A \in \mathcal{A} : A \cap \{T\wedge s \leq t\} \in \mathcal{F}_t \text{ for every } t \geq 0\}$$ (as usual). Show that $E[X_{T \wedge t} | \mathcal{F}_{T\wedge s}] = E[X_{T \wedge t} | \mathcal{F}_s]$ for every $0 \leq s \leq t$. (I should maybe add that the problem comes with the hint that, for any stopping time $S$, the random variable $X_S \textbf{1}_{S < \infty}$ is measurable with respect to $\mathcal{F}_S$.)
I have made the following observation so far: Since $T \wedge s \leq s$, we have $\mathcal{F}_{T \wedge s} \subset \mathcal{F}_{s}$. Thus, it would be sufficient to show that $E[X_{T \wedge t} | \mathcal{F}_s]$ is measurable with respect to the smaller sigma field $\mathcal{F}_{T \wedge s}$.
However, I have absolutely no idea how to tackle showing such a statement. Neither in general, nor in this particular case.
If any of you had some suggestions or hints on what I could try next, I would really appreciate it.