Dynkin ’ s Lemma in Measure Theory

For simplicity, we adopt the following rules: O1 denotes a non empty set, f denotes a sequence of subsets of O1, X, A, B denote subsets of O1, D denotes a non empty subset of 21 , n, m denote natural numbers, F denotes a non empty set, and x, Y denote sets. Next we state two propositions: (1) For every sequence f of subsets of O1 and for every x holds x ∈ rng f iff there exists n such that f(n) = x. (2) For every n holds PSegn is finite. Let us consider n. One can verify that PSegn is finite. Next we state the proposition (3) For all sets x, y, z such that x ⊆ y holds x misses z \ y. Let a, b, c be sets. The functor a, b followed by c is defined as follows: (Def. 1) a, b followed by c = (N 7−→ c)+·[0 7−→ a, 1 7−→ b].