McShane integrability vs Bochner integrability #
In this file we prove that any Bochner integrable function is McShane integrable (hence, it is
⊥ integrable) with the same integral. The proof is based on
Russel A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock.
integral, McShane integral, Bochner integral
The indicator function of a measurable set is McShane integrable with respect to any locally-finite measure.
f is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
f has integral
y on a box
I with respect to a locally finite measure
a.e. equal to
g has the same integral on
A simple function is McShane integrable w.r.t. any locally finite measure.
For a simple function, its McShane (or Henstock, or
⊥) box integral is equal to its
integral in the sense of
f : ℝⁿ → E is Bochner integrable w.r.t. a locally finite measure
μ on a rectangular box
I, then it is McShane integrable on
I with the same integral.