Manifolds 41 | Measurable Sets and Null Sets

Measurable sets and null sets are the missing pieces that make integration on manifolds work beyond a single coordinate chart. Once a subset’s image in Euclidean space is measurable (in the usual Lebesgue sense), its “generalized volume” can be assigned consistently, and if that image has Lebesgue measure zero, the subset can be ignored without changing any integral. This is what turns the manifold integral from a chart-local construction into a global tool.

The discussion starts with the familiar case of the 2-sphere S2. Because S2 is an orientable 2-dimensional manifold, it carries a canonical volume form that measures areas. Using spherical coordinates, a parameterization F maps an open domain U in (θ, φ)-space into R3 and onto the sphere. To compute the area of a subset, the integral of the manifold volume form Ω over the subset is rewritten as an integral in Euclidean coordinates: it becomes an integral of the pullback F*Ω over the parameter domain. In this setup, the pullback introduces a Jacobian factor equal to sin θ, reflecting how area elements scale under the parameterization.

A key practical point follows: the parameterization does not cover the entire sphere—only a line of points is missed. Despite that omission, the area integral still comes out to 4π, the total surface area of S2. The missing line is treated as a null set: it has Lebesgue measure zero in the relevant coordinate representation. Since null sets contribute nothing to the integral, the “gap” in coverage does not affect the result. This example motivates the formal definitions that come next.

With the sphere as motivation, the text fixes the notions of measurable and null sets for an orientable n-dimensional manifold equipped with a volume form. A set A is called measurable if, for any chart (U, φ), the intersection A ∩ U maps under the chart into a measurable subset of R^n (measurable with respect to Lebesgue measure). Similarly, A is a null set if, under the chart map, the corresponding intersection has Lebesgue measure zero. Crucially, these properties must hold regardless of which chart is used to test them, so the definitions are compatible with the manifold’s atlas.

These definitions then extend the integral. Inside a chart, the integral of Ω over a measurable set A is well-defined. Moreover, if two sets differ only by a null set, their integrals agree: the integral over A can be defined by subtracting the null set from A, and the result is unchanged. That is exactly how the sphere computation justifies ignoring the parameterization’s missing line.

The final takeaway is that integration is now robust for measurable subsets, even when they don’t fit neatly into one chart. The remaining challenge—handled in a later step—is building a general theory for sets that require multiple charts to cover, which demands technical work to ensure the global integral stays consistent.