I'm trying to show that $E[E[\ \cdot\mid \mathcal{F}_\sigma]\mid\mathcal{F}_\tau]=E[E[\ \cdot\mid \mathcal{F}_\tau]\mid\mathcal{F}_\sigma]$ for stopping times $\sigma$ and $\tau$, I've come to the following claim:
Let Y be an integrable, $\mathcal{F}_\sigma$-measurable random variable. Then, $E[Y\mid\mathcal{F}_\tau]$ is also $\mathcal{F}_\sigma$-measurable. I have (almost) no doubt that this is true, but can't see how to prove it. The statement is clear if we had some relation between the sigma algebras (like $\mathcal{F}_\tau\subset\mathcal{F}_\sigma$, but this is not given).
Is there a quick way to get the assertion?