Introduction to MEasure Theory for a Course in Ergodic Theory

Measure theory introduction.

1. Introduction to Ergodic Theory
Lecture I { Crash course in measure theory
Oliver Butterley, Irene Pasquinelli, Stefano Luzzatto, Lucia Simonelli, Davide
Ravotti
Summer School in Dynamics { ICTP { 2018
Lecture I (Measure Theory) Introduction to Ergodic Theory

2. Why do we care about measure theory?
Dynamical systemT:X!X
What can we say about typical orbits?
Push forward on measuresT(A) =(T
1
A) forAX?
Lecture I (Measure Theory) Introduction to Ergodic Theory

3. The goal
?
Associate to every subsetAR
n
a non-negative number(A) with the following
reasonable properties
((0;1)
n
) = 1
(
S
k
Ak) =
P
k
(Ak) whenAkare pairwise disjoint
(A)(B) whenAB
(x+A) =(A)
Why isn't this possible? What's the best that can be done? What happens when we
drop some requirements?
Lecture I (Measure Theory) Introduction to Ergodic Theory

4. The goal
?
Associate to every subsetAR
n
a non-negative number(A) with the following
reasonable properties
((0;1)
n
) = 1
(
S
k
Ak) =
P
k
(Ak) whenAkare pairwise disjoint
(A)(B) whenAB
(x+A) =(A)
Why isn't this possible? What's the best that can be done? What happens when we
drop some requirements?
Lecture I (Measure Theory) Introduction to Ergodic Theory

5. The goal
?
Associate to every subsetAR
n
a non-negative number(A) with the following
reasonable properties
((0;1)
n
) = 1
(
S
k
Ak) =
P
k
(Ak) whenAkare pairwise disjoint
(A)(B) whenAB
(x+A) =(A)
Why isn't this possible? What's the best that can be done? What happens when we
drop some requirements?
Lecture I (Measure Theory) Introduction to Ergodic Theory

6. Denition
A collectionAof subsets of a spaceXis called analgebraof subsets if
; 2 A
Ais closed under complements, i.e.,A
c
=XnA2 AwheneverA2 A
Ais closed under nite unions, i.e.,
S
N
k=1
Ak2 AwheneverA1; : : : ;AN2 A
Lecture I (Measure Theory) Introduction to Ergodic Theory

7. Denition
A collectionAof subsets of a spaceXis called a-algebraof subsets if
; 2 A
Ais closed under complements, i.e.,A
c
=XnA2 AwheneverA2 A
Ais closed under countable unions, i.e.,
S
1
k=1
Ak2 AwheneverA1;A2: : :2 A
Lecture I (Measure Theory) Introduction to Ergodic Theory

8. Denition
A collectionAof subsets of a spaceXis called a-algebraof subsets if
; 2 A
Ais closed under complements, i.e.,A
c
=XnA2 AwheneverA2 A
Ais closed under countable unions, i.e.,
S
1
k=1
Ak2 AwheneverA1;A2: : :2 A
Denition
IfSa collection of subsets ofX, we denote by(S) the smallest-algebra which
containsS.
Lecture I (Measure Theory) Introduction to Ergodic Theory

9. Example
LetX=Rand letAdenote the collection of all nite unions of subintervals
Example
LetX=Rand letAdenote the collection of all subsets ofR
Denition
LetXbe any topological space. TheBorel-algebrais dened to be the smallest
-algebra which contains all open subsets ofX
Lecture I (Measure Theory) Introduction to Ergodic Theory

10. Example
LetX=Rand letAdenote the collection of all nite unions of subintervals
Example
LetX=Rand letAdenote the collection of all subsets ofR
Denition
LetXbe any topological space. TheBorel-algebrais dened to be the smallest
-algebra which contains all open subsets ofX
Lecture I (Measure Theory) Introduction to Ergodic Theory

11. Example
LetX=Rand letAdenote the collection of all nite unions of subintervals
Example
LetX=Rand letAdenote the collection of all subsets ofR
Denition
LetXbe any topological space. TheBorel-algebrais dened to be the smallest
-algebra which contains all open subsets ofX
Lecture I (Measure Theory) Introduction to Ergodic Theory

12. Denition
Ameasurable space(X;A) is a spaceXtogether with a-algebraAof subsets
Denition
Let (X;A) be a measurable space. Ameasureis a function:A ![0;1] such
that
(;) = 0
IfA1;A2: : :2 Ais a countable collection of pairwise disjoint measurable sets then


1
[
k=1
Ak
!
=
1
X
k=1
(Ak)
Denition
If(X)<1the measure is said to benite. If(X) = 1 the measure is said to be a
probability measure.
Lecture I (Measure Theory) Introduction to Ergodic Theory

13. Denition
Ameasurable space(X;A) is a spaceXtogether with a-algebraAof subsets
Denition
Let (X;A) be a measurable space. Ameasureis a function:A ![0;1] such
that
(;) = 0
IfA1;A2: : :2 Ais a countable collection of pairwise disjoint measurable sets then


1
[
k=1
Ak
!
=
1
X
k=1
(Ak)
Denition
If(X)<1the measure is said to benite. If(X) = 1 the measure is said to be a
probability measure.
Lecture I (Measure Theory) Introduction to Ergodic Theory

14. Denition
Ameasurable space(X;A) is a spaceXtogether with a-algebraAof subsets
Denition
Let (X;A) be a measurable space. Ameasureis a function:A ![0;1] such
that
(;) = 0
IfA1;A2: : :2 Ais a countable collection of pairwise disjoint measurable sets then


1
[
k=1
Ak
!
=
1
X
k=1
(Ak)
Denition
If(X)<1the measure is said to benite. If(X) = 1 the measure is said to be a
probability measure.
Lecture I (Measure Theory) Introduction to Ergodic Theory

15. Example
LetX=R. Thedelta-measure at a pointa2Ris dened as
a(A) =
(
1 ifa2A
0 otherwise
Translation invariant?Denition
A measure space is said to becompleteif every subset of any zero measure set is
measurable
Lecture I (Measure Theory) Introduction to Ergodic Theory

16. Example
LetX=R. Thedelta-measure at a pointa2Ris dened as
a(A) =
(
1 ifa2A
0 otherwise
Translation invariant?Denition
A measure space is said to becompleteif every subset of any zero measure set is
measurable
Lecture I (Measure Theory) Introduction to Ergodic Theory

17. Example
LetX=R. Thedelta-measure at a pointa2Ris dened as
a(A) =
(
1 ifa2A
0 otherwise
Translation invariant?Denition
A measure space is said to becompleteif every subset of any zero measure set is
measurable
Lecture I (Measure Theory) Introduction to Ergodic Theory

18. Denition
IfARwe deneouter measureto be the quantity


(A) := inf
(
1
X
k=1
(bkak) :fIk= (ak;bk)g
k
is a set of intervals which coversA
)
Denition
A setARis said to beLebesgue measurableif, for everyER,


(E) =

(E\A) +(EnA)
Denition
For any Lebesgue measurable setARwe dene the lebesgue measure(A) =

(A)
Lecture I (Measure Theory) Introduction to Ergodic Theory

19. Denition
IfARwe deneouter measureto be the quantity


(A) := inf
(
1
X
k=1
(bkak) :fIk= (ak;bk)g
k
is a set of intervals which coversA
)
Denition
A setARis said to beLebesgue measurableif, for everyER,


(E) =

(E\A) +(EnA)
Denition
For any Lebesgue measurable setARwe dene the lebesgue measure(A) =

(A)
Lecture I (Measure Theory) Introduction to Ergodic Theory

20. Denition
IfARwe deneouter measureto be the quantity


(A) := inf
(
1
X
k=1
(bkak) :fIk= (ak;bk)g
k
is a set of intervals which coversA
)
Denition
A setARis said to beLebesgue measurableif, for everyER,


(E) =

(E\A) +(EnA)
Denition
For any Lebesgue measurable setARwe dene the lebesgue measure(A) =

(A)
Lecture I (Measure Theory) Introduction to Ergodic Theory

21. Exercise A
1LetX=Nand letA=fAX:AorA
c
is niteg. Dene
(A) =
(
1 ifAis nite
0 ifA
c
is nite
:
Is this function additive? Is it countable additive?
2Show that the collection of Lebesgue measurable sets is a-algebra
Lecture I (Measure Theory) Introduction to Ergodic Theory

22. Exercise A
1LetX=Nand letA=fAX:AorA
c
is niteg. Dene
(A) =
(
1 ifAis nite
0 ifA
c
is nite
:
Is this function additive? Is it countable additive?
2Show that the collection of Lebesgue measurable sets is a-algebra
Lecture I (Measure Theory) Introduction to Ergodic Theory

23. Theorem (Carateodory extension)
LetAbe an algebra of subsets ofX. If

:A ![0;1]satises


(;) = 0,

(X)<1
IfA1;A2: : :2 Ais a countable collection of pairwise disjoint measurable sets and
S
1
k=1
Ak2 Athen



1
[
k=1
Ak
!
=
1
X
k=1
(Ak):
Then there exists a unique measure:A ![0;1)on(A)the-algebra generated
byAwhich extends

.
Lecture I (Measure Theory) Introduction to Ergodic Theory

24. Exercise B
1Show that there are subsets ofRwhich are not Lebesgue measurable
2Show that there are Lebesgue measurable sets which are not Borel measurable
Lecture I (Measure Theory) Introduction to Ergodic Theory

25. Exercise B
1Show that there are subsets ofRwhich are not Lebesgue measurable
2Show that there are Lebesgue measurable sets which are not Borel measurable
Lecture I (Measure Theory) Introduction to Ergodic Theory

26. Exercise B
1Show that there are subsets ofRwhich are not Lebesgue measurable (hint:
consider an irrational circle rotation, choose a single point on each distinct orbit)
2Show that there are Lebesgue measurable sets which are not Borel measurable
Lecture I (Measure Theory) Introduction to Ergodic Theory

27. Exercise B
1Show that there are subsets ofRwhich are not Lebesgue measurable (hint:
consider an irrational circle rotation, choose a single point on each distinct orbit)
2Show that there are Lebesgue measurable sets which are not Borel measurable
(hint: recall the Cantor function, modify it to make it invertible, consider some
preimage)
Lecture I (Measure Theory) Introduction to Ergodic Theory