Signed measures

Abstract

Given a measure space (X, E, μ) and a function ρ∈ L1(X, μ), the so-called Lebesgue indefinite integral ν(E)=∫Eρdμ(E∈E) defines a σ -additive set function, that is, if E= ⋃ nEn, with En∈ E a family of disjoint sets, then ν(E) = ∑ nν(En). Therefore, when ρ≥ 0, ν is a finite measure on E satisfying E∈E&μ(E)=0⇒ν(E)=0.