Integral of measurable spaces

Solution 1:

An integral is a generalization of a weighted sum, but neither adding measurable spaces nor multiplying them with numbers are meaningful operations. Neither can we build integrals of topological spaces, filters, uniformities, etc.

Since the question is inspired by products of measurable spaces, what one can do is forming direct sums. The booklet Borel spaces by Rao and Rao contains two approaches. Let $(X_\lambda,\mathcal{X}_\lambda)_{\lambda\in\Lambda}$ be a family of measurable spaces. We can assume the underlying spaces to be disjoint and $X=\bigcup_\lambda X_\lambda$. The measurable sets of the direct sum are of the form $\bigcup_\lambda B_\lambda$ for some $(B_\lambda)\in\prod_\lambda \mathcal{X}_\lambda$. The weak direct sum has as its underlying $\sigma$-algebra the family $\sigma\big(\bigcup_\lambda\mathcal{X}_\lambda\big)$. The direct sum and the weak direct sum coincide if and only if $\Lambda$ is countable.