Borel and Continuous Systems of Measures

We study Borel systems and continuous systems of measures, with a focus on mapping properties: compositions, liftings, fibred products and disintegration. Parts of the theory we develop can be derived from known work in the literature, and in that sense this paper is of expository nature. However, we put the above notions in the spotlight and provide a self-contained, purely measure-theoretic, detailed and thorough investigation of their properties, and in that aspect our paper enhances and complements the existing literature. Our work constitutes part of the necessary theoretical framework for categorical constructions involving measured and topological groupoids with Haar systems, a line of research we pursue in separate papers.

💡 Research Summary

The paper provides a self‑contained, measure‑theoretic treatment of two closely related notions: Borel systems of measures (BSM) and continuous systems of measures (CSM). A BSM is defined for a Borel map (p\colon X\to Y) between standard Borel spaces; it assigns to each point (y\in Y) a Borel measure (\mu_y) supported on the fibre (p^{-1}(y)) in such a way that the map (y\mapsto\mu_y) is Borel‑measurable. A CSM refines this concept for continuous maps between topological spaces: each (\mu_y) must be a Radon measure and the functional (y\mapsto\int_X f,d\mu_y) must be continuous for every compactly supported continuous function (f). The authors first clarify the relationship between the two: every CSM is a BSM, but the converse requires additional regularity and σ‑finiteness hypotheses.

The core of the work is a systematic study of four fundamental operations on such systems—composition, lifting, fibre product, and disintegration—and the precise mapping properties each operation preserves.

1. Composition. Given BSM/CSM (\mu) for (p\colon X\to Y) and (\nu) for (q\colon Y\to Z), the composed system (\lambda) for (q\circ p) is defined by (\lambda_z(A)=\int_Y \nu_z(dy),\mu_y(A)). The authors prove that (\lambda) satisfies the same measurability (or continuity) requirements as the original systems, essentially a Fubini‑Tonelli type theorem for systems of measures. The proof carefully tracks σ‑finiteness and regularity at each stage, showing exactly when the push‑forward and pull‑back operations commute.

2. Lifting. When a Borel (or continuous) map (p\colon X\to Y) is extended to a larger space (\tilde X) with a map (\tilde p\colon\tilde X\to Y) that shares the same fibres, a BSM/CSM on (X) can be “lifted’’ to (\tilde X). The paper spells out the necessary conditions—typically that (\tilde X) is a measurable (or topological) completion of (X) and that (\tilde p) is a core‑map of (p). Under these hypotheses the lifted system retains the BSM/CSM properties, which is crucial for constructing Haar systems on groupoids after passing to completions.

3. Fibre product. For two maps (p_1\colon X_1\to Z) and (p_2\colon X_2\to Z) equipped with systems (\mu_1) and (\mu_2), the fibre product space (X_1\times_Z X_2) carries a natural system obtained by taking the tensor product of the fibrewise measures: ((\mu_1\otimes_Z\mu_2)z = \mu{1,z}\times\mu_{2,z}). The authors verify that σ‑finiteness, regularity, and continuity are preserved under this construction, and they prove a Fubini‑type identity for integrals over the fibre product. This result is the technical backbone for defining product Haar measures on the space of composable arrows in a groupoid.

4. Disintegration. The most delicate part of the paper deals with the converse of the push‑forward operation: given a measure (\nu) on (X) and a Borel map (p\colon X\to Y), one seeks a family ({\mu_y}{y\in Y}) such that (\nu = \int_Y \mu_y,d(p*\nu)(y)) and each (\mu_y) is supported on the fibre (p^{-1}(y)). Using the Radon–Nikodym theorem together with regularity assumptions, the authors establish existence and (almost everywhere) uniqueness of such a disintegration. They also give sufficient conditions—e.g., (p) open, (X) locally compact Hausdorff, and (\nu) Radon—for the resulting family to be a CSM. This theorem underpins the construction of Haar systems on measured groupoids, where one needs to split a global Haar measure into conditional measures on each source fibre.

After developing the abstract theory, the paper applies it to measured and topological groupoids. For a groupoid (\mathcal G) with source and target maps (s,t\colon\mathcal G\to\mathcal G^{(0)}), the authors show how a Haar system can be viewed as a CSM for (s) (or (t)), and how the four operations above guarantee that the system behaves well under groupoid multiplication, inversion, and restriction to units. The exposition is deliberately self‑contained: all needed lemmas about Radon measures, regularity, and Borel measurability are proved, and a summary table collects the hypotheses required for each operation.

In conclusion, the paper achieves three main goals. First, it isolates BSM and CSM as distinct but closely related objects and clarifies the precise bridge between them. Second, it provides a unified, rigorous treatment of composition, lifting, fibre product, and disintegration, filling gaps that are only implicitly addressed in the existing literature. Third, it demonstrates that this framework is exactly what is needed for categorical constructions involving measured groupoids and their Haar systems, thereby laying a solid foundation for the authors’ forthcoming work on groupoid C(^*)-algebras and related dynamical systems.