Integral of Real-Valued Measurable Function

For simplicity, we follow the rules: X denotes a non empty set, Y denotes a set, S denotes a σ-field of subsets of X, F denotes a function from N into S, f , g denote partial functions from X to R, A, B denote elements of S, r, s denote real numbers, a denotes a real number, and n denotes a natural number. Let X be a non empty set, let f be a partial function from X to R, and let a be a real number. The functor LE-dom(f, a) yields a subset of X and is defined as follows: (Def. 1) LE-dom(f, a) = LE-dom(R(f),R(a)). The following three propositions are true: (1) |R(f)| = R(|f |).