Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by $E=\bigcap_{n\geq 1} E_n$ where $$E_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k), x>0\right\}$$ is dense in $F$, assuming that $\displaystyle \lim_{n\rightarrow\infty}\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ for all $f$ in $C^2(\mathbb{R}_+^2;\mathbb{R})$, and where $g_n$ and $g$ are positively supported probability distribution functions such that $\frac{X_n}{n}$ converges in distribution to $X$ if $X_n\sim g_n$ and $X\sim g$.

The problem essentially comes from the interchange between the first and second components in the nonlocal boundary condition. More precisely, had the problem been to show that $G=\bigcap_{n\geq 1} G_n$ where: $$G_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(x,\frac{k}{n}\right)g_n(k), x>0\right\}$$ were dense in: $$H=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(x,z)g(z)dz, x>0\right\},$$ it would have sufficed to define the sequence of functions $f_n$ by: $$ f_n(x,y)=f(x,y)+\frac{y}{\sum_{j\in\mathbb{N}}\frac{j}{n}g_n(j)}\left(\int_\mathbb{R} f(x,z)g(z)dz-\sum_{j\in\mathbb{N}} f\left(x,\frac{j}{n}\right)g_n(j)\right)$$ to deduce that $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(x,\frac{k}{n}\right)g_n(k)=\int_\mathbb{R} f(x,z)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$, obviously with the inverted assumption that $\displaystyle \lim_{n\rightarrow\infty}\sum_{k\in\mathbb{N}} f\left(x,\frac{k}{n}\right)g_n(k)=\int_\mathbb{R} f(x,z)g(z)dz$.

Unfortunately, I have not been able to find a suitable sequence of functions to use in my initial problem where the components are interchanged in the boundary condition. Any ideas or references to literature would be greatly appreciated.

Update: As an example, the sequence of functions $f_n$ defined by: $$ f_n(x,y)=f(x,y)+\mathbb{1}_{\{y>0\}}\left(\int_\mathbb{R} f(z,y)g(z)dz-\sum_{j\in\mathbb{N}} f\left(\frac{j}{n},y\right)g_n(j)\right)$$ would verify $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$, but $f_n$ is discontinuous (I am looking for a function in $C^2(\mathbb{R}_+^2;\mathbb{R}))$.