Probability on Finite Set and Real-Valued Random Variables

One can prove the following four propositions: (1) Let X be a non empty set, S1 be a σ-field of subsets of X, M be a σ-measure on S1, f be a partial function from X to R, E be an element of S1, and a be a real number. Suppose f is integrable on M and E ⊆ dom f and M(E) < +∞ and for every element x of X such that x ∈ E holds a ≤ f(x). Then R(a) ·M(E) ≤ ∫ f E dM. (2) Let X be a non empty set, S1 be a σ-field of subsets of X, M be a σ-measure on S1, f be a partial function from X to R, E be an element of S1, and a be a real number. Suppose f is integrable on M and E ⊆ dom f and M(E) < +∞ and for every element x of X such that x ∈ E holds a ≤ f(x). Then R(a) ·M(E) ≤ ∫ f E dM.