Integration of compactly supported top forms #
A finite open chart cover is disjointed into Borel coordinate pieces. Integration over these pieces requires no Hausdorff assumption or measurable structure on the manifold. Integrability is expressed in every chart and every model basis, independently of a cover or an orientation.
For the mathematical construction of integration on oriented manifolds, see J. M. Lee, Introduction to Smooth Manifolds, Chapter 16. Here finite disjoint Borel chart pieces provide the localization, and their common refinement proves independence of the chosen cover.
A compactly supported top form is integrable when its unsigned coefficients are integrable in every preferred chart and every model basis. The model uses its Borel measurable structure.
Instances For
The coordinate characterization of integrability, with any Borel measurable structure on the model. Neither a chart cover nor an orientation occurs in this condition.
Integrability on an open chart cover introduces the intrinsic predicate for rough forms. One model basis suffices, and neither form continuity nor an orientation is required.
A compact rough form supported in a preferred chart is intrinsically integrable exactly when its single coefficient in one model basis is integrable.
The signed coefficient of an integrable top form in a preferred chart is integrable for the Haar measure of any basis, when the chart sign is continuous.
Integrable top forms have compact topological support.
Multiplication by a nonzero scalar preserves top-form integrability.
The sum of two integrable top forms is integrable.
The zero form is integrable in top degree. An explicit model basis also covers an empty manifold, where no nonzero form can certify the degree.
Multiplication by any constant scalar preserves top-form integrability.
Negation preserves top-form integrability.
Finite sums of integrable top forms are integrable.
The total integral of a rough top form against a manifold orientation. It is zero unless the form is compactly supported and integrable. A disjoint Borel chart cover defines the sum; the manifold needs no Hausdorff or measurable-space structure.
Instances For
A nonintegrable top form has zero total integral.
A form without compact support has zero total integral.
Any finite open chart cover computes the canonical integral through its disjoint Borel pieces. The formula is independent of the cover and model basis and permits overlapping charts.
A compact manifold admits a finite disjoint chartwise Borel partition computing every integrable top-form integral. No measurable structure on the manifold is installed.
The top-form integral is additive on integrable rough forms.
Reversing the manifold orientation reverses the total integral, including its value on nonintegrable rough forms.
The zero top form has zero integral.
Integration commutes with finite sums of integrable top forms.
Constant scalar multiplication commutes with the total top-form integral, even for nonintegrable rough forms.
A preferred chart computes the whole integral when the form vanishes outside that chart up to a chartwise null set. The premise uses the actual nonzero values, rather than their closure, so null faces or endpoints may be omitted from the chart.
An integrable form supported in a preferred chart has that chart's integral. This applies to rough forms, including discontinuous coefficients, and to any representing orientation lift.