# Probability Constructions

$y_n$ 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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One commonplace meaning of $y_y\to y$ in this context is that $\int x\;dy_n \to \int x\;dy$ for every bounded continuous function $x$. –  Michael Hardy Oct 1 '11 at 19:20
....and that suggests the $x$ in the examples should be either unbounded or discontinuous. I suspect they will need to be unbounded. –  Michael Hardy Oct 1 '11 at 19:22
@MichaelHardy I thought $\int x \mathrm{d} y$ referred to the mean. –  Sasha Oct 1 '11 at 20:03
@Sasha: Oh.... you mean as in $\int x \; dy(x)$. I was thinking of something like $\int_\mathbb{R} x(u) \; dy(u)$. –  Michael Hardy Oct 1 '11 at 23:59
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. –  Michael Hardy Oct 2 '11 at 0:19