8/8/2019 Probability Theory Lecture notes 06

1/4

Outline

Contents

1 Lebesgue-Stieltjes Measure and Distribution Functions 1

2 Measurable functions 3

1 Lebesgue-Stieltjes Measure and Distribution Func-

tions

Lebesgue measure review, the construction

Let F0 be the field generated by the collection of all intervals, 0 be the usuallength measure of intervals.

Extend 0 to 1 : G R, where G is F0 plus limiting sets ofF0, 1 on theselimiting sets are defined by exchange the order of limit and measure.

Extend 1 to , which is an outer measure defined on 2. Unfortunately, in

general does not satisfy countable-additivity.

Restrict to the collection of measurable sets, denoted by F, which is a -algebra.

Lebesgue measure review, the main results

The Carathodory extension theorem. There exists one and only one way toextend a -finite measure 0 on an algebra F0 to, F,the -algebra generated by

F0.

The measure approximation theorem. For any A F and a given > 0, thereexists a set B F0 such that (AB) < .

Generalizations

Slight generalization of Lebesgue Measure: ((a, b]) = F(b) F(a), whereF() is a non-decreasing, continuous function.

Further generalization: F() just needs to be a right-continuous function. So

jumps are allowed. Def. of right-continuity: F(xn) F(x) when xn x.

Such an F is called a Stieltjes measure function. If is a probability measure, itis called the distribution function of. We will use these two terms interchange-

ably.

8/8/2019 Probability Theory Lecture notes 06

2/4

Theorem (1.5), pg. 440. Given such an F, there is a measure s.t. ((a, b]) =F(b) F(a).

Definitions

The converse of Theorem (1.5) is almosttrue. For most measures we can define its

distribution function. The only exceptions: ((a, b]) = for some finite interval.

Definition 1. A Lebesgue-Stieltjes measure on R is a measure : B R such that(I) < for each boundedinterval.

Alternatively, we may define L-S measure by F() which satisfies the non-decreasingand right-continuity conditions.

Comments and examples

Page 1.4.5.

Discrete measure

Page 26.

Let be a L-S measure that is concentrated on a countable set S = {x1, x2, . . . , }.

Distribution function: step function.

can be extended to 2.

Restriction

In the discrete measure example, we can restrict to 2S instead ofB. It pretty

much has the same property.

Remark: why we dont say restrict on S?

Another restriction example: is concentrated on some interval [a, b].

Construction ofB[a, b]. Then we can restrict on this -field without loosingits mathematical properties.

L-S measure on Rn

Sketch of construction

The analogy of intervals: rectangles.

Open rectangles, closed rectangles, semi-closed rectangles: just need to knowtheir two vertices. Same notation: (a, b]

The smallest -field containing all rectangles: B(Rn).

A L-S measure on Rn is a measure : B(Rn) R such that (I) < foreach bounded rectangle I.

2

8/8/2019 Probability Theory Lecture notes 06

3/4

L-S Measure on Rn (II)

Its distribution function is defined to be F = ((, x]).

These distribution functions are also

non-decreasing. For a b, that is, a1 b1, a2 b2, . . ., we have: F(a) F(b).

Right-continuous. F is right continuous in all variables.

On the other hand, just as in the 1-dim case, for any non-decreasing, right con-tinuous function, there exist a unique measure on B(Rn) corresponding withit.

Measure of a finite rectangle

This is a main difference.

((a,b]) = F(b) F(a).

Draw a two dimensional example to illustrate this point.

Page 442-443 describes an elaborated way of measuring a rectangle by means ofthe distribution function, but that formula is not often used. The reason is that

later we will see that we can define density function for most common, useful

distributions, and there is a easy way to calculate measure of a rectangle (or an

arbitrary region for that matter) using integral.

2 Measurable functions

An example of measurable function

A probability example to show the motivation. Let (,F, P) be a probabil-ity space. To be more specific, lets say = {HEAD, TAIL-a, TAIL-b}, F ={, , {HEAD} , {TAIL-a, TAIL-b}}, ({HEAD}) = 1

2, ({TAIL-a, TAIL-b}) =

1

2.

is a probability measure. Interpretation: probability of seeing HEAD or TAIL(including a,b, types) is both 1/2.

Now define a function h on in this way: h(HEAD) = 1, h(TAIL-a) =h(TAIL-b) = 1. (Such a function is sometimes called a coding function).

An example of measurable function (II)

This function codes HEAD, TAIL-a, TAIL-b into numbers, which are much eas-ier to process than plain English descriptions!

One nice thing is, we can talk about P(h = 1) and P(h = 1) instead of(HEAD), ({TAIL-a, TAIL-b}).

3

8/8/2019 Probability Theory Lecture notes 06

4/4

h not only maps arbitrary events into tangible numbers, but also maps a measuredefined on an arbitrary space to a measure defined for numbers.

Another example (II)

More complex examples: measuring height of trees. : trees (descriptive). P:probability of the height of trees. h : R maps every tree to a real number(of centimeters, or inches, etc).

In this example, we may want to estimate probabilities in this form: P(a