Marginalization of the conditional expectation

Question

Let $Y, Z, X_1, X_2, T$ be random variables. Let $\mathbb{E}(Z\mid X_1, X_2) = \mathbb{E}[Y\mid T=t, X_1, X_2]$. I need a property $$\mathbb{E}(Z\mid X_1) = \mathbb{E}[Y\mid T=t, X_1].$$ Does it hold when $T\perp X_2\mid X_1$, where '$\perp$' means conditional independence? Does it hold even without this independence?

Answer 1

Within the ambiguity of talking about conditional expectation in a set of null measure ($T=t$) , we could say that equality is valid in case of conditional independence: \begin{gather*} \mathbb E[Z\mid X_1]=\int\mathbb E[Z\mid X_1,X_2=x]dP_{X_2/X_1}=\int\mathbb E[Z\mid X_1,X_2=x]dP_{X_2/T=t,X_1}=\\=\int\mathbb E[Y\mid T=t,X_1,X_2=x]dP_{X_2/T=t,X_1}=\mathbb E[Y\mid T=t,X_1] \end{gather*}