$E[E(ZX\mid Y)\mid Z,Y]$ can be simplified? Any comment would be much appreciated.
2026-04-04 12:07:13.1775304433
How do I simplify $E[E(ZX\mid Y)\mid Z,Y]$?
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$\mathbb E[ZX\mid Y]$ is by defintion $\sigma (Y)$ measurable, and thus $\sigma (Z,Y)$ measurable. Therefore, $$\mathbb E\Big[\mathbb E[ZX\mid Y]\mid Y,Z\Big]=\mathbb E[ZX\mid Y].$$