Can You Pass Nonlinear Functions of Conditioned Variable Through Conditional Expectation?

Question

In general, nonlinear functions cannot pass through the expectation operator. For example, it is not generally true that $E\left(e^X\right)=e^{E(X)}$ (we can only use Jensen's Inequality here).

However, when one conditions on $X$, is this true? Does it become true, for example, that $E\left(e^X\cdot Y|X\right)=e^XE(Y|X)$?

Thanks!

Edit: More generally, is it true that $E(X\cdot f(Y)|Y)=f(Y) E(X|Y)$ for a nonlinear function $f$?

Answer 1

More generally, is it true that $E(X\cdot f(Y)|Y)=f(Y) E(X|Y)$ for a nonlinear function $f$?

Yes, see David Williams, Probability with martingales, or any decent presentation of conditional expectation (say, pages 5-6 of this).