$y_n$ is a sequence of probability measures on $\mathbb{R}$ such that $y_n\rightarrow y$ where $y$ is another probability measure on $\mathbb{R}$.
Construct an example where:
$\int x \; dy_n$ exists for each $n$ and has a finite limit but $\int x \; dy$ is $+\infty$.
$\int x \; dy_n$ exists for each $n$ and $\lim_{n \to \infty }\int x \; dy_n=+\infty$, but $\int x \; dy$ is finite.