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$y_n$ 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:

  1. $\int x \; dy_n$ exists for each $n$ and has a finite limit but $\int x \; dy$ is $+\infty$.

  2. $\int x \; dy_n$ exists for each $n$ and $\lim_{n \to \infty }\int x \; dy_n=+\infty$, but $\int x \; dy$ is finite.

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    It appears I may have been understanding the notation in a way that is different from what was intended. So here is a rephrasing of my earlier comment. One commonplace meaning of $y_n\to y$ is that $\int g(x) \; dy_n(x) \to \int g(x) \; dy(x)$ for every bounded continuous function $g$. Another equivalent way is that the sequence of cumulative distribution functions corresponding to $y_n$ converges to the c.d.f. corresponding to $y$ except possibly at points where the latter is not continuous.2011-10-02

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