For what g, $E(X|g(X)=0$ where $X$ is standard normal?

Question

I know if $g(X)=X^2$, the equality holds. Is there any other $g$? How can we generalize the case $g(X)=X^2$ (other than $g(X)=X^{2k}$?

Answer 1

Let $Y=g(X)$ and let $h$ denote some measurable function such that $E(X\mid Y)=h(Y)$. Assume that $g$ is even. Then $Y=g(-X)$, hence, the distribution of $X$ being symmetric, $(-X,Y)$ and $(X,Y)$ are identically distributed. Conditional expectations only depends on distributions hence $E(-X\mid Y)=h(Y)$, that is, $h(Y)=0$.