Suppose I have three random variables $X_1,X_2, V$, and I want the following condition to hold:
$E[X_1^2|X_2<V<X_1,X_1,X_2]=E[X_1^2|X_1]=h(X_1)$,
i.e., I want conditioning varibles $V$ and $X_2$ to drop from the conditioning set. I'm wondering whether this equality holds by construction without the need for adding any additional assumptions. This is because given the value of $X_1$ in the conditioning set, $E[X_1|X_2<V<X_1,X_1,X_2]=X_1^2$ . We don't need conditions such as $X_1,X_2, V$ being mutually independent to deliver this, right?