I am reading an old paper from 1949 by Shapely, Karlin and Bohnenblust titled "Solutions of Discrete Two-persons Games"
They introduce the following definitions
However I am unsure how these definitions should be interpreted. It doesn't seem like $\Sigma$ represents a sum operation here as the sets $I_{1}(x)$ and $J_{2}(y)$ seem to be sets of indices, so I don't see a reason why you would want to add elements of the sets together. Should I interpret $\Sigma$ here as a union of the sets of indices instead? If so how should I interpret $\Pi$? Once again it doesn't seem like a standard product operator?
Looks like union to me. Each $I_1(x)$ is a set of indices, as you mention, and after applying the "$\Sigma$" operation, the result is used to test set inclusion.
In words, the operation appears to collect all of the indices of pure strategies which are active in any optimal mixed strategy.
I suspect the "product" $\Pi$ is product in the sense of sets, i.e. elements of its resultant are ordered tuples. Without seeing how $I_2$ is used later, that's only a guess.