Egoroff ’ s Theorem Noboru Endou

In this paper n, k are natural numbers, X is a non empty set, and S is a σ-field of subsets of X. Next we state several propositions: (1) Let M be a σ-measure on S, F be a function from N into S, and given n. Then {x ∈ X: ∧ k (n ≤ k ⇒ x ∈ F (k))} is an element of S. (2) Let F be a sequence of subsets of X and n be an element of N. Then (the superior set sequence of F )(n) = ⋃ rng(F ↑ n) and (the inferior set sequence of F )(n) = ⋂ rng(F ↑ n). (3) Let M be a σ-measure on S and F be a sequence of subsets of S. Then there exists a function G from N into S such that G = the inferior set sequence of F and M(lim inf F ) = sup rng(M ·G).