Inferring independence from conditional expectation

Question

Given two random variables $X,Y$ defined on the same probability space, if $E[X|Y]$ is not a function of $Y$ can we say that $X$ and $Y$ are independent?

Answer 1

No. It is permisable for $\mathsf E(X\mid Y)$ to be constant , but have $X$ dependent on $Y$.

You can surely imagine a distribution where the conditional expected value for $X$ given $Y$ is constant, but have the realised values for $X$ be restricted by values for $Y$.

Consider $(X,Y)\sim\mathcal U\{(0,1),(1,0),(2,1)\]$

Answer 2

We can always express $E(X|Y)$ as $f(Y)$ for some measurable function $f: \mathbb R \to \mathbb R$ (whether or not there is independence).