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Measure_Int

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Universidade Federal de Santa Catarina
Departamento de Matemática
Measure Theory and Integration
Matheus Cheque Bortolan
Florianópolis - SC
2018.2
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CONTENTS
1 Measure Spaces 5
1.1 σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Product σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Borel σ-algebras on topological spaces . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 The extended real numbers . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Elementary families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 σ-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Solved exercises from [1, Page 24] . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Measures 19
2.1 Basic notions and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Solved exercises from [1, Page 27] . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Outer measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Solved exercises from [1, Page 32]) . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 Borel measures on the real line . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5.1 The Lebesgue measure on the real line . . . . . . . . . . . . . . . . . 59
2.5.2 The Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.5.3 Cantor-type sets of positive measure . . . . . . . . . . . . . . . . . . 63
2.5.4 A set not Lebesgue measurable in R . . . . . . . . . . . . . . . . . . 64
2.6 Solved exercises from [1, Page 39] . . . . . . . . . . . . . . . . . . . . . . . . 65
3 Integration 73
3.1 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1.1 Decompositions of functions . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.2 Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1.3 Measurability of functions on complete measure spaces . . . . . . . . 80
3.2 Solved exercises from [1, Page 48] . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 Integration of nonnegative functions . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 Solved exercises from [1, Page 52] . . . . . . . . . . . . . . . . . . . . . . . . 95
3.5 Integration of complex functions . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.5.1 Comparison between the Riemann and Lebesgue integrals . . . . . . 106
3.5.2 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.6 Solved exercises from [1, Page 59] . . . . . . . . . . . . . . . . . . . . . . . . 112
3.7 Modes of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.8 Solved exercises from [1, Page 63] . . . . . . . . . . . . . . . . . . . . . . . . 129
3.9 Product measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.9.1 Monotone classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.9.2 The Fubini-Tonelli Theorem . . . . . . . . . . . . . . . . . . . . . . . 142
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3.10 Solved exercises from [1, Page 68] . . . . . . . . . . . . . . . . . . . . . . . . 146
3.11 The n-dimensional Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . 152
3.11.1 The Jordan content meets Lebesgue measure . . . . . . . . . . . . . . 156
3.11.2 The Change of Variables Theorem . . . . . . . . . . . . . . . . . . . . 159
3.12 Solved exercises from [1, Page 76] . . . . . . . . . . . . . . . . . . . . . . . . 167
4 Signed Measures and Differentiation 177
4.1 Signed measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.2 Solved exercises from [1, Page 88] . . . . . . . . . . . . . . . . . . . . . . . . 182
4.3 The Lebesgue-Radon-Nikodym Theorem . . . . . . . . . . . . . . . . . . . . 185
4.4 Solved exercises from [1, Page 92] . . . . . . . . . . . . . . . . . . . . . . . . 190
4.5 Complex measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
4.6 Solved exercises from [1, Page 94] . . . . . . . . . . . . . . . . . . . . . . . . 199
5 Lp Spaces 203
5.1 Basic theory of Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.2 Solved exercises from [1, Page 186] . . . . . . . . . . . . . . . . . . . . . . . 212
5.3 The dual of Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
5.4 Solved exercises from [1, Page 191] . . . . . . . . . . . . . . . . . . . . . . . 234
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ACKNOWLEDGMENTS
These notes were created fo provide an expanded version of a part of Folland’s book
(see [1]) on Measure Theory and Integration, and a little of Lp spaces. Here I prove many
results that are left for the reader, correct some small mistakes present in the book, as well as
provide solutions for many exercises. This is not an independently created material, and all
the credit of the theory and exercises is due to [1]. I do not claim ownership of any content
presented in these notes.
I would like to thank all the students of 2018.2 - Measure Theory and Integration.
Alessandra, Ben-Hur, Bruna, Daniella, Edivania, Elizangela, Hernán, João Paulo, João
Pering, Kledilson, Lucas, Marduck, Talles and Thais, thank you so much for your contribution
with the creation and revision of these notes, pointing out mistakes and suggesting easier
solutions for so many exercises. I hope these notes made the subject a little bit easier to
digest!
For whoever is using these notes: if you wish to suggest corrections and comments, please
e-mail me at m.bortolan@ufsc.br. Feel free to e-mail asking me for the .tex files of these
notes if you wish.
Happy measuring and integrating!
“The problem is not the problem. The problem is your attitude about the problem.”
Captain Jack Sparrow.
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CHAPTER 1
MEASURE SPACES
1.1 σ-ALGEBRAS
In this section X is a nonempty set, N = {1, 2, 3, · · · } represents the positive integers and
N0 = {0, 1, 2, 3, · · · } represents the nonnegative integers.
Definition 1.1.1. An algebra of sets in X is a nonempty collection A of subsets of X
such that given A,B ∈ A we have
A ∪B ∈ A and Ac = X \ A ∈ A.
In other words, A is an algebra if it is closed under unions and complements.
It is clear that if A is an algebra then given A,B ∈ A we have A ∩ B ∈ A, since
A ∩B = (Ac ∪Bc)c. Now if n is a fixed positive integer and E1, · · · , En ∈ A then
n⋃
i=1
Ei ∈ A and
n⋂
i=1
Ei ∈ A.
Moreover we have the following:
Proposition 1.1.2. If A is an algebra in X then ∅ ∈ A and X ∈ A.
Proof. Since A is nonempty, there exists a set A ∈ A. Therefore we have ∅ = A ∩ Ac ∈ A
and also X = ∅c ∈ A. �
Definition 1.1.3. A σ-algebra A in X is an algebra A which is closed under countable
unions, that is, if {En}n∈N ⊂ A and E =
∞⋃
n=1
En then E ∈ A.
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Clearly a σ-algebra A is also closed by countable intersections, since
⋂
n
En = (
⋃
n
Ecn)
c.
Remark 1.1.4. It is worth to point out that an algebra A is a σ-algebra if it is closed
under disjoint unions. In fact let {En}n∈N ⊂ A. Then given k ∈ N we have
Fk = Ek \
( k−1⋃
n=1
En
)
= Ek ∩
( k−1⋃
n=1
En
)
∈ A.
Hence the sequence {Fk}k∈N is in A, is pairwise disjoint and
∞⋃
n=1
En =
∞⋃
k=1
Fk ∈ A.
Example 1.1.5. Given a nonempty set X, are σ-algebras:
1. A = {∅, X}.
2. A = P(X), the collection of all subsets of X.
3. If X is uncountable:
A = {E ⊂ X : E is countable or Ec is countable}.
In fact, given A,B ∈ A then A ∪ B is countable if both A and B are countable and
(A∪B)c = Ac ∩Bc is countable if at least one of them has countable complement. Also
Ac ∈ A, thus A is an algebra. A similar argument shows that
⋃
n∈NEn ∈ A if En ∈ A
for all n ∈ N, and hence A is a σ-algebra.
This σ-algebra is called the σ-algebra of countable or co-countable sets.
Proposition 1.1.6. Let {Aλ}λ∈Λ be a collection of σ-algebras in X, indexed over a set
Λ. Then ⋂
λ∈Λ
Aλ = {E ⊂ X : E ∈ Aλ for all λ ∈ Λ}
is also a σ-algebra in X.
Proof. It is straightforward. �
Let E ⊂ P(X). Then, using Proposition