In short, I want to examine whether the expression $$E\left[g(x,y)|y\right]=\int_Xg(x,y)dF(x,y)$$ is differentiable in $y$.
Given an event space $X\subset\mathbb{R}$, the function $g:X\times \mathbb{R}\rightarrow \mathbb{R}$ is bounded, differentiable in $y$ and weakly monotone in $x$ and $y$. $F(x,y)$ is a cdf for the random variable and differentiable in $y$ (and weakly decreasing in $y$).
How can I show that the expression above is differentiable in $y$ for a general event space $X$? I have shown it for the special cases if $X$ is discrete or if $X$ convex with an existing pdf $f(x,y)$. However, I fail to show it for a general case.
I know that I have to show that the expression $$\lim\limits_{h\rightarrow0}\frac{\int_X g(x,y+h)dF(x,y+h)-\int_X g(x,y)dF(x,y)}{h} $$ exists. My first idea is to use the definition of the Riemann-Stieltjes integral over the upper sums to replace the expressions by $$\int\limits_X g(x,y)dF(x,y)=\lim\limits_{\text{mesh}(P)\rightarrow0}\sum\limits_{i=1}^n g(x_i,y)\left(F(x_i,y)-F(x_{i-1},y)\right)$$, where $P$ is the usual partition of $X$ into $n$ parts. However, I am not entirely comfortable by proceeding to switch the limit operators (although the two functions are continuous) and even if it works, I am not sure whether my resulting expression $$\int\limits_X g_y(x,y)dF(x,y)+\int\limits_X g(x,y)dF_y(x,y)$$ is well-defined (since the partial derivative $F_y$ is not increasing over $X$).
Any hint where I went wrong or any other idea how to proceed would be highly appreciated!