Basic theorems in measure theory can be presented in the following way. A Riesz space is an ordered Q-vector space, which is a lattice (it is then automatically distributive). It is sigma-complete iff any bounded sequence has a l.u.b. We consider only Riesz space with a strong unit 1: for any x there exists n such that -n 1 <= x <= n 1
Let X be a compact Hausdorff, C(X) the Riesz space of continuous function on X. One basic remark is that the Riesz space B(X) of bounded Baire functions on X is the sigma completion of C(X). In this way both the Riesz-Markov representation theorem and the monotone convergence theorem get a natural interpretation: integral of Baire functions (and measure of Baire subsets) can be directly defined by this universal property of B(X), using the next point.
Given a positive linear functional I on C(X), another basic remark is that I_f = I_f1 \/ I_f2 if f = f_1\/f_2 and I_f(g) = I(fg). The map f |--> I_f can hence be extended in a unique way to B(X). This gives directly the Riesz-Markov representation theorem -and- the monotone convergence theorem.
An attempt to present the early history of measure theory
With Erik Palmgren. We compare some constructive approaches to measure theory and give a pointfree presentation of the notion of Borel sets.
An early attempt to define measure on compact Haussdorf spaces A better approach is to work with ring or Riesz space of basic continuous functions, see Stone spectrum