Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$

Then I have seen in some place claiming the following:

for $h:\Bbb R \times S \to \Bbb R$ and bounded, continuous with $x_n \to x$.

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$

Then I have seen in some place claiming the following: $$\int h(x_n,y)\mu_n(dy) \to \int h(x,y)\mu(dy)$$

for $h:\Bbb R \times S \to \Bbb R$ and bounded, continuous with $x_n \to x$.

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$

Then I have seen in some place claiming the following: $$\int h(x_n,y)\mu_n(dy) \to \int h(x,y)\mu(dy)$$

for $h:\Bbb R \times S \to \Bbb R$ and bounded, continuous with $x_n \to x$.

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$

Then I have seen in some place claiming the following:

for $h:\Bbb R \times S \to \Bbb R$ and bounded, continuous with $x_n \to x$.