The Lebesgue Monotone Convergence Theorem

For simplicity, we adopt the following rules: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, E is an element of S, F , G are sequences of partial functions from X into R, I is a sequence of extended reals, f , g are partial functions from X to R, s1, s2, s3 are sequences of extended reals, p is an extended real number, n, m are natural numbers, x is an element of X, and z, D are sets. Next we state a number of propositions: (1) If f is without +∞ and g is without +∞, then dom(f + g) = dom f ∩ dom g. (2) If f is without +∞ and g is without −∞, then dom(f − g) = dom f ∩ dom g.