5 - Integration

Summary

For a locally BV set E, let R(E) := D[CH_*(E))}. It follows from Theorem 3.6.6 the derivation D: CH_*(E) → R(E) is bijective, and we shall investigate its inverse I_E: R(E) → CH_*(E). We show that I_E, which has properties analogous to those of the indefinite Lebesgue integral, can be applied to partial derivatives of pointwise Lipschitz functions, and we prove unrestricted versions of the Gauss-Green and Stokes theorems. We also show that an averaging process akin to the classical Riemann integral provides a direct definition of I_E.

The R-integral

In this section we define the R-integral and prove some of its basic properties; most of them follow readily from the corresponding properties of AC* charges established in Section 3.6.

Definition 5.1.1. Let E be a locally BV set. A function f defined on E is called R-integrable in E if there is an F ∈ CH_*(E), called the indefinite R-integral of f, such that DF(x) = f(x) for almost all x ∈ E.

The family of all R-integrable functions in a locally BV set E is denoted by R(E). It follows from Theorem 3.6.6 that the indefinite R-integral F of a function f ∈ R(E) is uniquely determined by f, and we denote it by (R) ∫ f dL^m or (R) ∫ f(x) dx.