Let $G$ be a $\sigma$-field.
Is the statement: $$ \mathbb{E}[\mathbb{E}[X|G]] = \mathbb{E}[X] $$ true even if X is NOT a G-random variable?
2026-04-02 03:12:26.1775099546
Expectation of conditional expectation
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Not sure what $G$-random variable means in this context. If you are saying that $X$ is independent of $G$, then $$ \mathbb{E}[X|G] = \mathbb{E}[X], $$ which implies $$ \mathbb{E}[\mathbb{E}[X|G]] = \mathbb{E}[\mathbb{E}[X]] = \mathbb{E}[X]. $$