I have the following problem:
Let $X,Y$ be random variables with distributions $P_X,P_Y$ and $f_0$ be a map from the support of X,Y to the reals. I define a new function $\chi_0(y) = E_X[f_0(X,y)]$. If I have a function $f$ that is close to $f_0$ in the $L_2(X,Y)$ sense, i.e.
$\lvert\lvert f-f_0\rvert\rvert_{L_2(X,Y)}^2 = E_{X,Y}\left[\left(f(X,Y)-f_0(X,Y)\right)^2\right]\leq \epsilon$
What can I say about the function $\chi_f(y)=E_X\left[f(X,y)\right]$? I am interested in saying something about either $\lvert\lvert \chi_f-\chi_0\rvert\rvert_{L_2(Y)}^2$ or $\lvert\chi_f(y)-\chi_0(y)\rvert^2$.
For the first I thought about using Jensen's inequality and the following:\begin{align}\lvert\lvert \chi_f-\chi_0\rvert\rvert_{L_2(Y)}^2&=\int\left(\chi_f(y)-\chi_0(y) \right)^2dP_Y\\\\&=\int\left(\int f(x,y)-f_0(x,y)dP_X \right)^2dP_Y\\\\&\leq\int\left(\int f(x,y)-f_0(x,y) \right)^2dP_XdP_Y\end{align}I could bound this to obtain the desired result if $dP_XdP_Y\leq dP_{XY}$. Is this reasonable?
For the second:\begin{align}\lvert\chi_f(y)-\chi_0(y)\rvert^2&=\left\lvert\int f(x,y)-f_0(x,y)dP_X\right\rvert^2\\\\&\leq\int f(x,y)-f_0(x,y)^2dP_X \\\\&=\lvert\lvert f(x,y)-f_0(x,y)\rvert\rvert_{L_2(X)}^2\end{align}
I am not sure how to continue, if possible. Moreover, how does this change if $f$ is multivariate $(X_1,\dots,X_n)$ and $\chi(x_n)=E_{X_1,\dots,X_n}\left[f(X_1,\dots,x_n)\right]?$ Thank you in advance