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.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- chapter 4 probability and counting rules
- probability and statistics
- probability formula review
- sample space events and probability
- probability handout the center for brains minds machines
- lecture notes on probability and statistics eusebius
- axiomatic probability and point sets
- 1 probability conditional probability and bayes formula
- lecture notes measure theory and probability
- math 2p82 mathematical statistics lecture notes
Related searches
- strategic management lecture notes pdf
- financial management lecture notes pdf
- business management lecture notes pdf
- organic chemistry lecture notes pdf
- corporate finance lecture notes pdf
- philosophy of education lecture notes slideshare
- business administration lecture notes pdf
- advanced microeconomics lecture notes pdf
- microeconomics lecture notes pdf
- statistics and probability notes pdf
- probability theory and statistics
- marketing lecture notes pdf