When does the expectation of a random variable X depend on the order in which its support is listed? Reading Blitzstein and Hwang’s introduction to probability http://probabilitybook.net (really good so far!).
On page 150, they mention in regard to the definition of the expectation of a discrete random variable that
The expectation of a discrete random variable X is undefined if the value of the expectation depends on the order in which the support is listed.
I understand that if we the expectation varies depending on said order then we of course don’t have a well-defined expectation(i.e. a single value).
My question(s):

*

*What do we mean by the order in which the support is listed?

*When(some example as well please) does the expectation depend on this ordering?

Thanks :)

This is a general mathematical result that $\mathbb E[X]$ is only defined when either (i) $$\mathbb E[|X|] < \infty$$or (ii) $$\mathbb E[|X|] = +\infty$$and one of the expectations $$\mathbb E[\underbrace{X~\mathbb I_{X>0}}_{X^+}]\qquad\mathbb E[\underbrace{X~\mathbb I_{X<0}}_{-X^-}]$$is finite. (Note that $\mathbb E[|X|]=\mathbb E[X~\mathbb I_{X>0}]-\mathbb E[X~\mathbb I_{X<0}]$ since $X=X^+-X^-$ and $|X|=X^++X^-$.)
The remark about the ordering is connected with the Riemann series theorem. Quoting from Wikipedia:

As an example, the series 1 – 1 + 1/2 – 1/2 + 1/3 – 1/3 + ...
converges to 0 (for a sufficiently large number of terms, the partial
sum gets arbitrarily near to 0); but replacing all terms with their
absolute values gives 1 + 1 + 1/2 + 1/2 + 1/3 + 1/3 + ... , which sums
to infinity. Thus the original series is conditionally convergent, and
can be rearranged (by taking the first two positive terms followed by
the first negative term, followed by the next two positive terms and
then the next negative term, etc.) to give a series that converges to
a different sum: 1 + 1/2 – 1 + 1/3 + 1/4 – 1/2 + ... = ln 2. More
generally, using this procedure with p positives followed by q
negatives gives the sum ln(p/q). Other rearrangements give other
finite sums or do not converge to any sum.

