Given two random variables Y and Z, I want to show that
$$ Var(Y|Z) = 0 $$
if and only if there exists a measurable function $h$ such that
$$ Y = h(Z).$$
From $Var(Y|Z) = 0$ I know that $E[Y^2|Z] = E[Y|Z]^2$, but I don't know what to make of this or how to continue from here.
Note
$$ Var[Y|Z] = E[(Y-E[Y|Z])^2|Z] = 0$$
Since $(Y - E[Y|Z])^2 \geq 0$, its expectation equals to $0$ if and only if $(Y - E[Y|Z])^2 = 0$ almost surely, i.e. $Y = E[Y|Z]$ almost surely. By definition of conditional expectation, $E[Y|Z]$ is a $\sigma(Z)$-measurable function. And the last step of the claim should be similar to this:
