probability theory lecture notes 04

8/8/2019 Probability Theory Lecture notes 04

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Outline

Contents

1 Measures 1

1 Measures

Countable additivity of a set function

Let () be an extended real-valued set function defined on a -fieldF, i.e.,

It takes sets A in F as input variables.

It maps these sets into extended real numbers, e.g., real numbers plus .

It is said to be

finitely additive if: For finite collection ofdisjointsets A1, . . . , AN inF, N

n An

=

n (An).

Countably additive (-additive) if: For countably infinite collection of disjointsets A1, . . . , An, . . . in F,

n

An

=n

(An). (1)

Basic definition

A measure on a -field F is a function (A) such that

It is non-negative: (A) 0.

Measure of an empty set is always zero: () = 0. (Remark: The reverse is nottrue.)

It is countably additive.

Exercise: finite/countable additivity are about commuting union and addition of dis-

jointsets. What happens ifA1, A2, . . . are not disjoint?

Finite/Probability Measure

If() is finite, is called a finite measure.

Let 2(A) = C1(A). 1, 2 share a lot of mathematical properties.

As a special case, for every finite measure , we can definite (A) = 1()(A)

so that () = 1.


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If() = 1, is called a probability measure.

In other words, essentially every finite measure is equivalent to a probabilitymeasure.

By convention, we define a measure/probability space to be a triple (,F, ). Where is the whole space, F is the -field with as its whole space, and is a measuredefined on F.

About

The term of a probability space is a set of all outcomes, and are not necessarilynumbers.

For example, we sometimes denote the set of two possible outcomes of a Bernoullidistribution as = {H, T}, represents Head and Tail.

In fact you can have defined as the set of humans such asall students who aretaking BST401, a -algebra containing all subsets of you, and define a proba-

bility to quantify the probability that one of you guys in such a subset will one

day win the Nobel prize.

Examples

General definition of a counting measure: for any , is well defined on thelargest -field 2.

Counting measures on a finite set, Z, and Q.

Formally prove that every counting measure is a valid measure.

Discrete probabilities: K unique outcomes: = {1, . . . , K}, p1, . . . , pK ,Kk=1pk = 1, (A) =

kA

pk Here K can be countably infinite.

Bernoulli distribution. (fair coin/non-fair coin).

Binomial distribution. = {0, 1, . . . , N }, F = 2 pk =Nk

pk(1

p)Nk.

Poisson distribution. = Z+ = {0, 1, 2, . . .}, F = 2Z+

, pk = e

k

k! .

Basic Properties of a Measure

Monotonicity. IfA B, then (A) (B).

Subadditivity. If A iAi, then (A)

i (Ai). As a special case,(iAi)

i (Ai).

Continuity from below. Ai A = (Ai) (A).

Continuity from above. Ai A = (Ai) (A).

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The Borel -fields

Borel measure on R: = R, B, (A) = the length of A.

Where the Borel -field is defined to be the smallest -field containing all openintervals (a, b). (or all closed intervals [a, b], or all intervals of form (a, b].)

First step: open sets. More details will be discussed later.

Terminology: the smallest -field containing a collection of sets S is called a)the minimal -field over S; b) the -field generated by S.

B is a -field generated by the collection of open/closed intervals/sets.

Not every subset ofR is a Borel set! Checkout the Vitali set from Wikipedia.

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probability theory lecture notes 04

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