Conditional Expectation of product, where one conditional expectation is already known.

Question

Suppose we have random variables $X,Y,Z$ in a probability space $(\Omega, \mathcal{F}, P)$ and a sigma sub algebra $\mathcal{G} \subset \mathcal{F}$. Suppose it holds that $E[X|\mathcal{G}] = Y$. Does it hold that

$E[XZ| \mathcal{G}] = E[YZ|\mathcal{G}] $.

By the tower property of conditional expectation, it holds for sure, if $Z$ is $\mathcal{G}$ measurable. But in general I don't know.

Answer 1

Short answer No. Take $Z = X$ and non-$\mathcal G$ measurable then,

$$\mathbb E\left[XZ\Big | \mathcal G\right] = \mathbb E \left[X^2 \Big |\mathcal G\right] \neq \mathbb E \left[X \Big |\mathcal G\right]^2 = Y^2 = \mathbb E\left[YZ\Big | \mathcal G\right]$$