In this article, equation 2.2 is defined by
$$ \text{Var}(\mathbb{E}[Y|X,Z]) - \text{Var}(\mathbb E[Y|Z]) $$ and is said to be positive, unless $\mathbb{E}[Y|X,Z] = \mathbb{E}[Y|Z]$ almost surely. The only assumption here is that the expectations exist, there are no assumptions about the random variables $X,Z,Y$.
Is it obvious why this quantity is non-negative? Could someone please explain why, and if possible provide a reference to some readings to help better understand what is going on here?
Write $\theta=E(Y|X, Z)$, so $E(Y|Z)=E(\theta|Z)$. Since $E(\theta)=E(E(Y|Z))=E(Y)$, we see $$ \text{Var}(\theta)=E(\theta^2)-E(Y)^2, \ \ \ \ \text{Var}(E(Y|Z))=E(E(Y|Z)^2)-E(Y)^2. $$ By conditional Jensen's inequality, $$ \Big(E(\theta|Z)\Big)^2\leq E(\theta^2|Z), $$ Integrate this we see $$ E(E(Y|Z)^2)\leq E(\theta^2). $$ That proves the inequality.