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I am trying to improve a rather technical proof. In the old version of the proof, I indexed a finite, ordered sequence of unique positive real numbers $a_1, \dotsc, a_n$. I figured out that using these numbers directly as indices considerably streamlines the notation and further increases readability (as these numbers and not their indices are what matter). For example, instead of

$$ \sum_{\substack{j=1\\a_j<b}}^n a_j \sum_{i∈I_j} c_i, $$

I would now write (with everything being clear in context):

$$ \sum_{a<b} a \sum_{i∈I_a} c_i.$$

The only “problem” is that that I am using a real number as an index in expressions like $I_a$ (and a function notation is not feasible here) and also perform an induction over these numbers. This is maybe a bit of mental gymnastics, but I am confident that it is the lesser evil and mathematically sound.

Now, my question is this: I am probably not the first one to do such a thing. Is there a name for this notational strategy or something similar that I could use as a reference? The goal is to provide further reading for the interested or skeptical reader. It may help that in my context, one can also think of the $a_i$ as equivalence classes.

I found this question and indexed families, but neither satisfies me.

Wrzlprmft
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  • @CalumGilhooley: but I'm afraid I don't have time to look it up right now. – Here is some bounty to motivate you. – Wrzlprmft Jul 31 '20 at 14:02
  • For what it's worth, this sort of thing is done all the time in combinatorics with no comment or guilt. Just minutes ago in an email I wrote $\sum_{T \in \mathrm{SYT}(\lambda)} q^{\mathrm{maj}(T)} t^{\mathrm{des}(T)}$. Here the sum is over some combinatorial objects, and one can think of $\lambda$ as the "index" even though it's an integer partition instead of an integer. Sometimes it's helpful to write $\sum_{a : a<b}$ instead of $\sum_{a < b}$. – Joshua P. Swanson Jul 31 '20 at 20:14
  • I'm not sure this rises to an answer: I would just do it without much explanation - sums indexed by arbitrary finite sets are commonplace and even infinite sets are used sometimes (with an appropriate definition - probably involving absolute convergence or convergence of nets). Induction might raise some eyebrows - but you can either use indexes there or, to avoid that, point out that your set is well-ordered by $<$ (because it is finite), meaning you can induct on it directly (or you can induct on a natural number such as "this is true so long as no more than $n$ terms are less than $b$") – Milo Brandt Aug 05 '20 at 16:06

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As has been pointed out in a comment, people do this sort of thing all the time. I have only been able to find two texts that can be used as references for the topic. Of the two, the more readable is Terence Tao, Analysis I (1st edition, Hindustan Book Agency 2006). Here are some extracts:

Definition 7.1.6 (Summations over finite sets). Let $X$ be a finite set with $n$ elements (where $n \in \mathbf{N}$), and let $f \colon X \to \mathbf{R}$ be a function from $X$ to the real numbers (i.e., $f$ assigns a real number $f(x)$ to each element $x$ of $X$). Then we can define the finite sum $\sum_{x \in X} f(x)$ as follows. We first select any bijection $g$ from $\{i \in \mathbf{N} : 1 \leq i \leq n\}$ to $X$; such a bijection exists since $X$ is assumed to have $n$ elements. We then define $$ \sum_{x \in X} f(x) := \sum_{i=1}^n f(g(i)). $$

[$\ldots$]

To verify that this definition actually does give a single, well-defined value to $\sum_{x \in X} f(x),$ one has to check that different bijections $g$ from $\{i \in \mathbf{N} : 1 \leq i \leq n\}$ to $X$ give the same sum. In other words, we must prove

Proposition 7.1.8 (Finite summations are well-defined). Let $X$ be a finite set with $n$ elements (where $n \in \mathbf{N}$), let $f \colon X \to \mathbf{R}$ be a function, and let $g \colon \{i \in \mathbf{N} : 1 \leq i \leq n\} \to X$ and $h \colon \{i \in \mathbf{N} : 1 \leq i \leq n\} \to X$ be bijections. Then we have $$ \sum_{i=1}^n f(g(i)) = \sum_{i=1}^n f(h(i)). $$

The only other useful reference I know of is Nicolas Bourbaki, Elements of Mathematics: Algebra I, Chapters 1–3 (Hermann, Paris 1974, PDF). I’ll only quote one of his results, but the book might be worth exploring further, if it is important to prove things about sums of families defined on finite ordered sets without relying, as Tao does, on the special case where the indices are integers. (I haven’t tried to reproduce Bourbaki’s typography exactly, because I don’t think it’s possible in MathJax, and it would even require some hacking about in LaTeX.)

Theorem 1 (Associativity theorem). Let $E$ be an associative magma whose law is denoted by $\odot.$ Let $A$ be a totally ordered non-empty finite set, which is the union of an ordered sequence of non-empty subsets $(B_i)_{i \in I}$ such that the relations $\alpha \in B_i,$ $\beta \in B_j,$ $i < j$ imply $\alpha < \beta$; let $(x_\alpha)_{\alpha \in A}$ be an ordered sequence of elements of $E$ with $A$ as indexing set. Then $$ \bigodot_{\alpha \in A} x_\alpha = \bigodot_{i \in I}\Big(\bigodot_{\alpha \in B_i} x_\alpha\Big). $$

Wrzlprmft
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Calum Gilhooley
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  • Thank you for your answer. I removed many asides that seem to add nothing in terms of answering my question (and also are more extensive than what copyright laws allow for such purposes). I honestly have no idea why you went through the trouble of including all that. – Wrzlprmft Aug 08 '20 at 06:51
  • I did wonder what copyright law allowed. Do you have a reference that explains the law, so that I don't make the same (exhausting!) mistake again? I wouldn't call those passages "asides", though. The matter of copyright aside, all the results were germane. I wanted my answer to be a useful self-contained reference, giving an idea of what was in the book, but omitting proofs. Here I restore the link to my own question on the subject: Finite unordered sums. – Calum Gilhooley Aug 08 '20 at 10:12
  • Do you have a reference that explains the law, so that I don't make the same (exhausting!) mistake again? – There is no definite answer and it depends a bit on the specific jurisdiction. In general, we rely on the right to quote when quoting and this is limited to reasonable amounts. What exactly is reasonable depends on the context and would be decided by a court in a specific case. – Wrzlprmft Aug 08 '20 at 10:34
  • Mind that a good way to avoid copyright issues and keep the text concise is summarising portions for example like: “Tao continues to make and prove utility propositions such as … and …” From the asker’s point of view (i.e., mine), it is nice to know that these things are there, but I certainly do not need them expanded in all detail – after all, you don’t need to convince me; I just need material to convince others. – Wrzlprmft Aug 08 '20 at 10:36
  • Bear in mind that Stack Exchange answers are intended to be read by more than just the OP. – Calum Gilhooley Aug 08 '20 at 12:13