How to write a compact list terms through set notation? I would like to ask a question about a mathematical notation used in economics, because I'm not an expert on it. I'm studying an economic model in which I understood measure theory is used, but I'm not sure because it has not been told. There are definitions in which there are summations for every element of a particular set.
For example, the definition of the composite good, one of the goods which are consumed in the model, which has n regions, is the following, in the slides
$$M_{r}=\Big [\sum_{s\in R} \int_{i \in N_s} [q_{sr}(j)]^{(\sigma -1)/\sigma} dj\Big ]^{\sigma/(\sigma-1)}$$
r, is the region in which the good is consumed, s is the region in which the good is produced, j is the variety of the good, R is the set of regions, $N_s$ is the mass of firms - I know there is a continuum of firms - and I have a question: if I want to write all the composite goods consumed in all regions, can I write under the M  $r \in R $ and under the summation sign $s,r \in R$? I mean, am I writing all the composite goods with this notation, or am I doing an exception with mathematics just to get to the point?
I'd like just to write all the R composite goods in the correct way.
Thanks in advance
 A: Personally, I'd write $$\sum\limits_{r \in R}M_r =\sum\limits_{r \in R}\left[\sum_{s \in R}\int\limits_{i \in N_s}[q_{sr}(j)]^{(\sigma - 1)/\sigma}dj\right]^{\sigma/(\sigma - 1)}$$
but I don't know if there's some other standard notation used in the field - it may well be that there's some other name given to this thing (and if I was writing it a lot, I'd probably come up with a name for it just to avoid having to deal with $\sum_{r\in R}M_r$ every time).
However, there are a couple of things which I can say for certain:

*

*$M_{r\in R}$ doesn't work unless you define it first: what does that mean? How are we combining all of these things? I don't know, unless you tell me (unless this is standard in the field, in which case just follow the standard).

*Combining the two sums definitely doesn't work, even if you define it: this is a problem of mathematics, rather than notation: if we throw out the extraneous part by  assuming $\sigma/(\sigma - 1) = 2$ and that $R$ only has two elements (neither of these actually matter - the problem appears regardless, except in a handful of essentially silly cases), then what you're after is something of the form

$$(a + b)^2 + (c + d)^2$$ (where $a$ and $b$ are the values of the two integrals with $r$ equal to our first value, and $c$ and $d$ the other two).
However, if you write the sums combined, what you've written down is instead $$(a + b + c + d)^2,$$ which is not the same thing at all (for example, if $a = b = c = 1$, then the first gives us $2^2 + 2^2 = 8$, whereas the second gives us $4^2 = 16$).
