Is this *really* a categorical approach to *integration*?

Here’s an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it’s called, “A Categorical Approach to Integration“.

The Abstract:

“We present a general treatment of measures and integrals in certain (monoidal
closed) categories. Under appropriate conditions the integral can be defined by a universal property, and the universal measure is at the same time a universal multiplicative measure. In the multiplicative case this assignment is right adjoint to the formation of the Boolean algebra of idempotents. Now coproduct preservation yields an approach to product measures.”

The Problem:

I’d like to find a way to use category theory to define or think about integration, at least over $\mathbb{R}^k$, ideally in some pragmatic fashion, without borrowing too heavily from some other theory of integration. So before I invest lots more time & effort than usual trying to understand the thing . . .

Does the pdf (or whatever) acheive anything like this? Does its “integration” really mean integration (like “area under the curve” and so on) or is it a false friend, as in “integral domain”?

Please excuse my ignorance. I am trying . . .

NB: Yeah, it does seem to be talking about integration, but let’s go a little deeper there if possible. My first question is now highlighted. It’s still open. I’ve thrown in the soft-question tag for good measure . . .

Solutions Collecting From Web of "Is this *really* a categorical approach to *integration*?"