The First Mean Value Theorem for Integrals

For simplicity, we use the following convention: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to R, and E is an element of S. One can prove the following three propositions: (1) If for every element x of X such that x ∈ dom f holds f(x) ≤ g(x), then g − f is non-negative. (2) For every set Y and for every partial function f from X to R and for every real number r holds (r f) Y = r (f Y ). (3) Suppose f is integrable on M and g is integrable on M and g−f is nonnegative. Then there exists an element E of S such that E = dom f∩dom g and ∫ f E dM ≤ ∫ g E dM.