I was wondering recently if there were any other systems of numbers besides the reals which you could do measure theory on. In the sense that, measure theory is formulated with measures from sets to the reals; why not sets to some other space?
To get anything resembling typical measure theory, it seems clear you need addition on the space, and this operation to be commutative and associative. We also want a least-upper-bound property on space, which means we need an ordering. If we take only these requirements, then sets like .. the integers, work, but this seems lacking in terms of topological structure: I want a space that feels "dense". The notion of https://en.wikipedia.org/wiki/Linear_continuum seems adequate here. So I'm curious,
What linear continua are there that can be equipped with an order-respecting commutative semi-group operation (called addition), possessing the least upper bound property?
The classic example being the nonnegative reals. There's the classic result that if we want to make a whole field out of the space, then they are the unique such object (the unique ordered field with the LUB property). I can see a case for wanting a space that additionally has a multiplication operation, so that we get product measures, but this seems perhaps as a secondary question.