If $X=E[X\mid Y]$ what can we say about $X$ and $Y$.

Question

Suppose that we have two random variables $X,Y \in L^1$ such that: $$ X=E[X\mid Y] $$ what can we say about relationship between $X$ and $Y$?

Is $X$ and $Y$ independent?

Is $X$ and $Y$ uncorrelated?

There is converse(loosely speaking) of this statement that say if $X$, $Y$ are independent than $E[X\mid Y]=E[X]$. So in my case $X$ would be a constant?

Answer 1

Since $E(X\mid Y=y)=\int x p(x\mid y) dx\implies\ E(X\mid Y)=g(Y)$ for some function $g$. Thus $X=E(X\mid Y)\implies X=g(Y)$. As an example, if $X=Y,\ E(X\mid Y)=Y=X$