Suppose that $X$ and $Y$ are random variables defined on $(\Omega, \mathcal{F}, \mathbb{P})$, and let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}$. The tower property of conditional expectation says that $\mathbb{E}[XY] = \mathbb{E}[\mathbb{E}[XY | \mathcal{G}]]$. Suppose that we do not impose any further assumptions on independence and measurability of $X$ and $Y$, I want to ask is it true that
\begin{equation}
\mathbb{E}[XY] = \mathbb{E}[X \mathbb{E}[Y | \mathcal{G}]],
\end{equation}
and how can one prove it if it is true. Any ideas?
Best Answer
No. Let $\mathcal G:=\{\varnothing,\Omega\}$.
Then $\mathbb E[Y\mid\mathcal G]=\mathbb EY$ so that: $$\mathbb E[X\mathbb E[Y\mid\mathcal G]]=\mathbb E[X\mathbb EY]=\mathbb EX\mathbb EY$$
This does not necessarily equal $\mathbb EXY$.