Reference for series on arbitrary infinite sets

Question

In class my professor introduced the following terminology:

Let $J$ be an arbitrary infinite set and let $f:J\to\mathbb{R}$ be a real-valued function. Then the symbol $\sum_{j\in J}f(j)$ is the series determined by $J$, with terms $f(j)$. We say that $\sum_{j\in J}f(j)$ converges to $S$ unconditionally if for every $\epsilon>0$, there is a finite subset $F_{0}\subset J$ such that if $F$ is any finite subset of $J$ that contains $F_{0}$, then

$\bigg|\sum_{j\in F}f(j)-S\bigg|<\epsilon$.

Does anyone know of any textbooks which discuss this topic? I haven't seen it in Rudin's Principles of Mathematical Analysis or in Royden and Fitzpatrick's Real Analysis. Any suggestions are appreciated!

Answer 1

It seems that this topic reappears on this site quite regularly and there are many questions and answers that discuss this type of sum. However, posts including references to texts discussing this definition of sum are less frequent (and if they are some references, they are mostly scattered across the site).

This is the reason why I have started this community wiki answer which collects some references for such sums. (So feel free to add further references you are aware if - that is actually the point of community wiki posts.)

It should also be added that in case we are working with non-negative functions, the definition can be stated in much simpler way - we can define $$\sum_{j\in J} f(j) = \sup\{\sum_{j\in F} f(j); F\text{ is finite}\}.$$ Since in some context this is entirely sufficient, some authors only deal with such functions and include this as the definition. This is the reason why I have divided the list into two parts.

References for non-negative functions

• G. Folland: Real Analysis, in Section 0.5. This book was also mentioned in this answer.
• T. Tao: Analysis I, in Section 8.2 named Summation on infinite sets.
• T. Tao: An Introduction to Measure Theory - in the introductory chapter. This book was also mentioned in this answer.

References for arbitrary functions

• Chapter IX of J. Dixmier: General Topology is devoted to definition and basic properties of these sums, the author studies them in context of arbitrary normed space.
• P. R. Halmos: "Introduction to Hilbert Spaces and the Theory of Spectral Multiplicity, in Section I.7. Halmos' book was mentioned as a reference in this answer.
• Christopher Heil: A Basis Theory Primer, in Section 3.2 (Convergence with Respect to the Directed Set of Finite Subsets of $N$)
• Chapter 13 in S. Roman: Advanced Linear algebra. This is discussed in the last part (section "The Arbitrary Case") and the sum is defined in the context of inner product spaces.

Answer 2

It may not be easy to find! Most likely your professor will eventually get synchronized with the assigned textbook, clearing up your questions.

You might find material on this site sufficient to put this in perspective. For a similar definition and discussion see this answer to the MSE question,

$\qquad$ Can we add an uncountable number of positive elements, and can this sum be finite?

So just go with the flow and keep studying! Consider asking your professor if there is some advantage to defining a series in this non-conventional manner. I would also want to know if the definition means that $S$, when it exists, must be unique (the wording seems to suggests that that is the intention).

Answer 3

This is something I've been wondering myself long time ago. I've first encountered this topic while taking a Measure Theory course with a french professor (btw, one of the best courses I've ever had, absolutely brilliant)

I asked him for references but he didn't say anything useful (I'm not sure if he didn't want to tell me or if he didn't know either) Anyways, I've come to realize that this is something which is often taught in france. If you can read french, then you can easily find several documents on the subject, just look for "familles sommables"

You can look at the french Wikipedia entry

• Famille sommable

There is a useful Master thesis, and this one is great too. Here are some exercises with solutions

If you want an english reference, have a look at Choquet's Topology. Apparently it appears in Dieudonné's treatise too, but I don't know where.

Finally, this is one of the most important things you want to know about this

Bonne lecture!