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I have started a table of contents measure theory - contents and started adding it as a floating toc to the relevant entries
Query archived from measure space:
Eric: Is there some nice “arrow theoretic” way to state the above? It seems to be screaming to be a functor or an internalization or something.
Eric: The “products” $\bigcup$ and $\sum$ look like they should be “products” in two different categories. Is that silly?
John Baez: $\bigcup$ is less like a “product” than a “sum” — also known as a coproduct. The collection of subsets of $X$ is a poset, which is a kind of category, and the union of a bunch of subsets can be seen as their coproduct in this category. Unfortunately I don’t see a great way to understand the sum of real numbers as a coproduct! So, I can’t quite do what I think you’re hoping for.
Toby: Well, they are still operations that make both $\Sigma$ and $\mathbf{R}^+$ into monoidal categories; since they're also posets, this makes them monoidal posets. We can talk about countable additivity using transfinite composition. I doubt that this adds much to the theory of measure spaces, but it points the way to some of the generalisation below, as well to possibilities for categorification.
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