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- StudyBlue
- United-kingdom
- University of Bristol
- Mathematics
- Mathematics 20008
- Yu
- Ch1: Introduction

Harry W.

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12
The σ-field

F is a σ-field of subsets of Ω, if it satisfies these axioms:

- Ø∈F
- if A
_{1},A_{2},...∈F, then ∪^{∞}_{i=1}A_{i}∈F - if A∈F, then A
^{c}∈F, where A^{c}:=Ω\A.

A σ-field G is a sub σ-field of the σ-field F if G⊂F.

Collection of Events Generating σ-field

The σ-field generated by a collection of events C, σ(C), is the smallest σ-field that contains C.

i.e. intersections of all σ-field containing C.

The Probability Triple

A probability space is a triple (Ω,F,P).

- Ω is the sample space, i.i. the set of all possible outcomes.
- F is a σ-field (also called σ-algebra) of subsets of Ω.

Usually, we drop the "subsets of Ω" if there is no confusion about what the sample space is.

The Power Set of S

The power set of S, written 2^{S} or {0,1}^{S}, is a set that consists of all subsets of S.

Probability Measure

A probability measure, P is a function P:F→[0,1] satisfying these axioms:

- P(Ø) = 0, P(Ω) = 1.
- (σ-additivity) if A
_{1},A_{2},...∈F are pair-wise disjoint i.e. A_{i}∩A_{j}=Ø ∀i,j, s.t i≠j, then

P(∪^{∞}_{i=1}A_{i}) = ∑^{∞}_{i=1}P(A)

Probability Space is a Function

A random variable X on a probability space (Ω,F,P) is a function X:Ω→R that maps

outcomes in F (i.e. ω∈Ω) to values in R.

Filtration

Stochastic Process

Markov Chain

Markov's Process

Increasing and Decreasing Sequences

Continuity of Probability

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