LECTURE NOTES MEASURE THEORY and PROBABILITY

LECTURE NOTES

MEASURE THEORY and PROBABILITY

Rodrigo Ban~uelos Department of Mathematics

Purdue University West Lafayette, IN 47907

June 20, 2003

2

I SIGMA ALGEBRAS AND MEASURES

?1 ?Algebras: Definitions and Notation.

We use to denote an abstract space. That is, a collection of objects called points. These points are denoted by . We use the standard notation: For A, B , we denote A B their union, A B their intersection, Ac the complement of A, A\B = A - B = {x A: x B} = A Bc and AB = (A\B) (B\A). If A1 A2, . . . and A = n=1An, we will write An A. If A1 A2 . . . and A = n=1An, we will write An A. Recall that (nAn)c = nAcn and (nAn)c = nAcn. With this notation we see that An A Acn Ac and An A Acn Ac. If A1, . . . , An , we can write

nj=1Aj = A1 (Ac1 A2) (Ac1 Ac2 A3) . . . (Ac1 . . . Acn-1 An), (1.1)

which is a disjoint union of sets. In fact, this can be done for infinitely many sets:

n=1An = n=1(Ac1 . . . Acn-1 An).

(1.2)

If An , then

nj=1Aj = A1 (A2\A1) (A3\A2) . . . (An\An-1).

(1.3)

Two sets which play an important role in studying convergence questions are:

limAn) = lim sup An =

Ak

n

n=1 k=n

(1.4)

and

limAn

=

lim inf

n

An

=

Ak .

n=1 k=n

(1.5)

3

Notice

c

(limAn)c =

An

n=1 k=n

c

=

Ak

n=1 k=n

=

Ack = limAcn

n=1 k=n

Also, x limAn if and only if x Ak for all n. Equivalently, for all n there is

k=n

at least one k > n such that x Ak0 . That is, x An for infinitely many n. For

this reason when x limAn we say that x belongs to infinitely many of the Ans

and write this as x An i.o. If x limAn this means that x Ak for some

k=n

n or equivalently, x Ak for all k > n. For this reason when x limAn we say

that x An, eventually. We will see connections to limxk, limxk, where {xk} is a

sequence of points later.

Definition 1.1. Let F be a collection of subsets of . F is called a field (algebra)

if F and F is closed under complementation and finite union. That is,

(i) F

(ii) A F Ac F

n

(ii) A1, A2, . . . An F Aj F .

j=1

If in addition, (iii) can be replaced by countable unions, that is if

(iv) A1, . . . An, . . . F Aj F ,

j=1

then F is called a ?algebra or often also a ?field.

Here are three simple examples of ?algebras. (i) F = {, },

4

(ii) F = {all subsets of },

(iii) If A , F = {, , A, Ac}.

An example of an algebra which is not a ?algebra is given by the following.

Let = R, the real numbers and take F to be the collection of all finite disjoint

unions of intervals of the form (a, b] = {x: a < x b}, - a < b < . By

convention we also count (a, ) as right?semiclosed. F is an algebra but not a

?algebra. Set

1

An

=

(0,

1

-

]. n

Then,

An = (0, 1) F .

n=1

The convention is important here because (a, b]c = (b, ) (-, a].

Remark 1.1. We will refer to the pair (, F) as a measurable space. The reason for this will become clear in the next section when we introduce measures.

Definition 1.2. Given any collection A of subsets of , let (A) be the smallest ?algebra containing A. That is if F is another ?algebra and A F, then (A) F.

Is there such a ?algebra? The answer is, of course, yes. In fact, (A) = F

where the intersection is take over all the ?algebras containing the collection A. This collection is not empty since A all subsets of which is a ?algebra. We call (A) the ?algebra generated by A. If F0 is an algebra, we often write (F0) = F 0.

Example 1.1. A = {A}, A . Then (A) = {, A, Ac, }.

5

Problem 1.1. Let A be a collection of subsets of and A . Set A A = {B A: B A}. Assume (A) = F. Show that (A A) = F A, relative to A.

Definition 1.2. Let = R and B0 the field of right?semiclosed intervals. Then (B0) = B is called the Borel ?algebra of R.

Problem 1.2. Prove that every open set in R is the countable union of right ?semiclosed intervals.

Problem 1.3. Prove that every open set is in B.

Problem 1.4. Prove that B = ({all open intervals}).

Remark 1.2. The above construction works equally in Rd where we take B0 to be the family of all intervals of the form

(a1, b1] ? . . . (ad, bd], - ai < bi < .

?2. Measures.

Definition 2.1. Let (, F) be a measurable space. By a measure on this space we mean a function ? : F [0, ] with the properties

(i) ?() = 0

and

(ii) if Aj F are disjoint then

? Aj = ?(Aj).

j=1

j=1

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download