Conditional expectations as functions

Question

Consider the following conditional expectation: $$ \mathbb{E}[f(X,Y,Z)\mid X,Y] $$ I know that it can be written as $m(X,Y)$, where $$ m(x,y)=\mathbb{E}[f(X,Y,Z)\mid X=x, Y=y]. $$ Is its section $y\mapsto m(X,y)$ equivalent to $$ y\mapsto \mathbb{E}[f(X,y,Z)\mid X]? $$

Answer 1

I don´t think so. What is true is the following:

When $Z$ is independent of $X$ and $Y$ then by this property of conditional expectations, $$ \mathbb E[f(X,Y,Z)|X,Y]=\mathbb E[f(x,y,Z)]\Big|_{x=X,y=Y}\,. $$ Likewise, when $X$ and $Z$ are independent of $Y$ then $$ \mathbb E[f(X,Y,Z)|Y]=\mathbb E[f(X,y,Z)]\Big|_{y=Y}\,. $$