Integrals in analysis and category theory

Question

Are integrals in analysis special cases of coends in category theory? They are both seen as weighted sums, denoted by $\displaystyle\int $ and share the same formal properties (for example Fubini's Theorem). Also notice that there is a very beautiful reformulation of the Yoneda Lemma $$F(a) = \int^{p \in I} \mathrm{Hom}(a,p) \otimes F(p)$$ for functors $F : I^{op} \to \mathcal{C}$ into a cocomplete category $\mathcal{C}$ which looks like the equation $$\mu(A) = \int_{p \in I} \chi_A(p) ~ d \mu(p)$$ for a measure $\mu$ on a measurable space $I$.

Notice that this question is not as silly as it seems: Limits in analysis can be seen as limits in category theory, see MO/9951 or Wikipedia (Topological limits).

If the measurable space $I$ is discrete, the question is about (infinite) series - are they special cases of coends?

Answer 1

Not an answer, but it was too long for a comment.

Lawvere "Metric spaces" paper relates the inner and outer measure on a probability space to Kan extensions (which can be written as ends/coends). This is p. 162:

As an example of the last corollary [see the paper], we could take for $X$ the space of nonnegative step functions of a probability space $S$ and for $Y$ the space of all nonnegative functions with the natural sup metric, and consider as $\varphi$ the elementary integral; thus in general we might call $y\in Y$ $\varphi$-integrable if [a certain right Kan extension, the inner measure] = [a certain left Kan extension, the outer measure].