How do I simplify $E[E(ZX\mid Y)\mid Z,Y]$?

Question

$E[E(ZX\mid Y)\mid Z,Y]$ can be simplified? Any comment would be much appreciated.

Answer 1

$\mathbb E[ZX\mid Y]$ is by defintion $\sigma (Y)$ measurable, and thus $\sigma (Z,Y)$ measurable. Therefore, $$\mathbb E\Big[\mathbb E[ZX\mid Y]\mid Y,Z\Big]=\mathbb E[ZX\mid Y].$$