Measures and integrals

SECTION 1 introduces a method for constructing a measure by inner approximation, starting from a set function defined on a lattice of sets. SECTION 2 defines a “tightness” property, which ensures that a set function has an extension to a finitely additive measure on a field determined by the class of approximating sets. SECTION 3 defines a “sigma-smoothness” property, which ensures that a tight set function has an extension to a countably additive measure on a sigma-field. SECTION 4 shows how to extend a tight, sigma-smooth set function from a lattice to its closure under countable intersections. SECTION 5 constructs Lebesgue measure on Euclidean space. SECTION 6 proves a general form of the Riesz representation theorem, which expresses linear functionals on cones of functions as integrals with respect to countably additive measures.