Show $\lim\limits_{n\to\infty}\mathbf E(f(X_n)g(Y))=\mathbf E(f(X)g(Y))$ 
Let $X_n,X,Y$ be real valued random variables, defined on the same space. Assume that $$\lim\limits_{n\to\infty}\mathbf E(f(X_n)g(Y))=\mathbf E(f(X)g(Y))$$ for all $f$, continuous and bounded and for all $g$ bounded and Borel. Prove that $(X_n,Y)$ converges in law to $(X,Y)$

Is $\displaystyle\mathbf E(f(X_n)g(Y))=\int\int f(x)g(y)P^{X_n}(dx)P^{Y}(dy)$ ?
and $P^{X_n}(dx)$ and $P^{Y}(dy)$ are equal, since they're defined on the same probability space $(\Omega,\mathcal A,P)$ ?
 A: We have to prove that for each continuous and bounded function $\psi\colon\mathbf R^2\to\mathbf R$, we have 
$$\tag{*}\lim_{n\to\infty}\mathbb E[\psi(X_n,Y)]=\mathbb E[\psi(X,Y)].$$
We first prove that we can reduce to continuous functions with compact support. Then, for such functions, we can approximate by finite sums of functions involved in the assumption of the text, as suggested by Siméon.
Assume that we proved $(*)$ for $\psi$ continuous with compact support. Then consider for a fixed $R$ a continuous function $\eta_R$ which takes the value $1$ on the ball of center $0$ and radius $R$, and vanishes outside the ball centered at the origin and radius $R+1$. Then $\eta_R\psi$ is continuous with compact support. Noticing that $\lim_{R\to \infty}\sup_n\mathbb E[(1-\phi_R)(X_n,Y)]=0$, we obtain that (*) holds for each continuous and bounded function.
Let $\psi$ be a continuous function with compact support. We can assume that it is contained in $[-R,R]^2$ for some $R$. By the Stone-Weierstrass theorem, we may approximate uniformly on $[-(R+1),R+1]$ the function $\psi$ by functions of the form $\psi_N:=\sum_{i=1}^Nf_i(x)g_i(y)$, $x,y\in\mathbf R$. Taking $h_R$ the one-dimensional equivalent of $\eta_R$ (except that we want it to vanish outside $[-(R+\delta),R+\delta]$ for a small $\delta$) and replacing $f_i$ by $f_ih_R$ and similarly for $g_i$, we obtain that the approximation takes place on $[-R,R]^2\cup(\mathbb R^2\setminus[-(R+\delta),R+\delta]^2)$. We thus obtain 
$$|\mathbb E[\psi(X_n,Y)-\psi(X,Y)]|\leqslant |\mathbb E[\psi_N(X_n,Y)-\psi_N(X,Y)|+|\mathbb E[\psi(X_n,Y)-\psi_N(X_n,Y)]|+|\mathbb E[\psi(X,Y)-\psi_N(X,Y)]|.$$
Since 
$|\mathbb E[\psi(X_n,Y)-\psi_N(X_n,Y)]|\leqslant 
\sup_{(x,y)\in [-R,R]^2\cup(\mathbb R^2\setminus[-(R+\delta),R+\delta]^2)}|\psi(x,y)-\psi_N(x,y)|+2\max\{\lVert \psi\rVert_\infty,\lVert \psi_N\rVert_\infty\}(\mathbb P\{X_n\in [R,R+\delta]\cup [-R-\delta,-R]\})$
we obtain by the assumption of the text that for each $N$,
$$\limsup_{n\to \infty}|\mathbb E[\psi(X_n,Y)-\psi(X,Y)]|\leqslant\sup_{(x,y)\in [-R,R]^2\cup(\mathbb R^2\setminus[-(R+\delta),R+\delta]^2)}|\psi(x,y)-\psi_N(x,y)|+2\max\{\lVert \psi\rVert_\infty,\lVert \psi_N\rVert_\infty\}\limsup_{n\to \infty}(\mathbb P\{X_n\in [R,R+\delta]\cup [-R-\delta,-R]\}+\mathbb P\{X\in [R,R+\delta]\cup [-R-\delta,-R]\}).$$
Using the fact that $X_n\to X$ in distribution (take $g=1$ in the assumption of the text) and assuming that $\pm R,\pm (R+\delta)$ are continuity points of the cdf of $X$, we obtain for such $R$ and $\delta$ that 
$$\limsup_{n\to \infty}|\mathbb E[\psi(X_n,Y)-\psi(X,Y)]|\leqslant\sup_{(x,y)\in [-R,R]^2\cup(\mathbb R^2\setminus[-(R+\delta),R+\delta]^2)}|\psi(x,y)-\psi_N(x,y)|\\+4\max\{\lVert \psi\rVert_\infty,\lVert \psi_N\rVert_\infty\}\mathbb P\{|X|\in [R,R+\delta]\}.$$
Since $\lVert \phi_N\rVert_\infty=\sup_{|x|,|y|\leqslant R}|\phi_N(x,y)|\leqslant \sup_{|x|,|y|\leqslant R}|\phi_N(x,y)-\phi(x,y)|+\lVert \psi\rVert_\infty$, the sequence $(\lVert\psi_N\rVert)_\infty$ is bounded and the conclusion follows.
