1. The problem statement, all variables and given/known data Let g: ℝ → ℝ be Lebesgue measurable, let μ(A) be a measure of Lebesgue measurable sets defined by μ(A) = ∫A f dx, where f: ℝ → ℝ is non-negative, with finite integral on compact intervals. Prove that ∫ℝ g(x)dμ(x) exists if and only if ∫ℝ g(x)f(x) dx exists, in which case the two are equal 2. Relevant equations N/A 3. The attempt at a solution I'm reviewing the definitions/construction of Lebesgue integrals but that is not leading to anything fruitful. I have that the first integral exists iff the integral of |g| exists, which exists iff the sup of all integrals of simple, bounded, measurable, finitely supported functions less than |g| is finite ... and then I am stuck. I suppose that seeing the equation ∫|g|dμ = sup Σai μ(Ai) = sup Σai ∫Ai f dμ (aiχAi being some simple function approximating |g|) causes me to "get" the reason for the equality -- if we imagine the Ai partitioning ℝ and getting smaller and smaller (small enough to encompass some single point x) we see that the a_i ≈ g(x) and that the integral ≈ f(x)dx. But actually showing the truth of this rigorously is an entirely different matter. Please advise me! Thanks.