The Measurability of Extended Real Valued Functions

For simplicity, we adopt the following rules: X is a non empty set, x is an element of X, f , g are partial functions from X to R, S is a σ-field of subsets of X, F is a function from Q into S, p is a rational number, r is a real number, n, m are natural numbers, and A, B are elements of S. Let us consider X and let us consider f . We say that f is finite if and only if: (Def. 1) For every x such that x ∈ dom f holds |f(x)| < +∞. Next we state three propositions: (1) f = 1 f. (2) For all f , g, A such that f is finite or g is finite holds dom(f + g) = dom f ∩ dom g and dom(f − g) = dom f ∩ dom g.