Generalization of the Riemann integral to functions on the sphere

Question

I want to do something a bit strange: define the Riemann/Darboux integral for bounded real functions defined on the n-dimensional sphere.

The catch: I cannot use Lebesgue integration theory to do this. This is because I want to do something similar to averaging the values of the function over an equidistributed sequence of points, and then claiming the averages tend to the integral as I take more and more points. This idea is true for Riemann-integrable functions (see Theorem 1.1 and Corollary 1.1 in the book Uniform Distribution of Sequences by Kuipers and Niederreiter) but it fails for Lebesgue-integrable functions.

In particular, a Dirichlet-type function (nonzero on a countable dense set and zero elsewhere) cannot be Riemann-integrable for my needs.

So, to avoid reinventing the wheel, I'm looking for a reference where someone went through defining the Riemann or Darboux integral for either a sphere or a more general space. The definition must allow the following theorems:

1. Continuous functions are Riemann-integrable.
2. Riemann-integrable functions are Lebesgue-integrable and the two integrals are the same.

Answer 1

You can find it in most of books an analysis of functions of several variables. Let me mention Apostol, Bartle, Marsden & Hofman and Spivak.