Brownian Motion and Stochastic Calculus
1 Preliminaries of Measure Theory Denition 1 F P is said to be an algebra if (1) 2 F (2) A;B 2 F implies A S B 2 F (3) A 2 F implies AC 2 F F is said to be a semialgebra or semi ring is (1) 2 F (2) A;B 2 F implies AB 2 F (3) A;B 2 F implies AnB Pn k=1Ak for some Ak 2 F Denition 2 Class: If A;B 2 implies AB 2 Denition 3 Class: if (1) 2 (2) A;B 2 with A B implies A B 2 (3) An 2 An " A implies A 2 Denition 4 Let L R be such that 2 L implies 1 2 L. L class: (1)1 2 L (2) L is linearly closed (3) n 2 L; 0 n " bounded or 2 L implies 2 L Proposition 5 (i)M being Class Class impliesM is algebra (ii) If Class contains the Class then