Some Notation
Functional Analysis
Definition
Lecture 5: The Lebesgue Measure
Let 2R denote the collection of all subsets of R.
Definition
Bengt Ove Turesson
For A ⊂ R and x ∈ R, let A + x denote the set {a + x : a ∈ A}.
Definition
September 13, 2015
For an interval I ⊂ R, let `(I ) denote the length of I .
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Measures
The Outer Lebesgue Measure
Definition
Problem
The outer Lebesgue measure m∗ (A) of a subset A of R is defined
by
X
∞
∞
[
m∗ (A) = inf
`((aj , bj )) : A ⊂
(aj , bj ) .
Does there exist a function µ : 2R → [0, ∞] with the following
properties:
(i) µ(I ) = `(I ) for every interval I ;
j=1
(ii) µ is translation invariant: µ(A + x) = µ(A) for any set
A ∈ 2R and any number x ∈ R;
j=1
Remark
R
(iii) µ is countably additive: If AS
1 , A2 , . . . ∈ 2P is a sequence of
∞
pairwise disjoint sets, then µ j=1 Aj = ∞
j=1 µ(Aj ).
Notice that the outer measure of a set may be infinite.
Notice
also that the definition of m∗ (A) makes sense since
S∞
A ⊂ j=1 (−j, j).
If bj < aj for some j, we interpret (aj , bj ) as ∅. Having all but
a finite number of intervals of this form in a covering of A
corresponds to finite coverings.
Such a function µ is called a measure.
Theorem (Vitali’s Theorem, 1905)
There does not exist a measure with the properties mentioned
above.
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The Outer Lebesgue Measure
The Outer Lebesgue Measure
Proposition
Proof: End
For any interval I , m∗ (I ) = `(I ).
Next suppose that I is a bounded interval.
Given ε > 0, there exists a closed interval J ⊂ I such that
`(J) > `(I ) − ε.
Proof: Beginning
Suppose first that I = [a, b].
Then
Let ε > 0. Since I ⊂ (a − ε, b + ε),
m∗ (I ) ≤ `((a − ε, b + ε)) = b − a + 2ε. This shows that m∗ (I ) ≤ b − a.
To
to show that
P∞prove the reverse inequality, it suffices
∞
j=1 `(Ij ) ≥ b − a for any covering (Ij )j=1 of I with open intervals. By
compactness, one can assume that the coverings are finite.
Given a finite covering (Ij )nj=1 of I , let x1 ≤ x2 ≤ . . . ≤ x2n be the
endpoints of the intervals, where x1 < a and b < x2n . Then
n
X
j=1
`(Ij ) ≥
`(I ) − ε < `(J) = m∗ (J) ≤ m∗ (I ) ≤ m∗ (I ) = `(I ) = `(I ).
Hence, m∗ (I ) = `(I ).
Suppose finally that I is unbounded.
For any number r > 0, there exists a compact interval J ⊂ I
such that `(J) = r .
2n−1
X
j=1
(xj+1 − xj ) = x2n − x1 > b − a.
Then m∗ (I ) ≥ m∗ (J) = `(J) = r .
This implies that m∗ (I ) = ∞.
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Sets with Measure Zero
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Properties of the Outer Lebesgue Measure
Definition
A subset A of R is said to have measure zero if m∗ (A) = 0.
Proposition
Proposition
Any countable subset of R has measure zero.
The outer Lebesgue measure has the following properties:
(i) m∗ (∅) = 0;
Proof.
(ii) if A ⊂ B, then m∗ (A) ≤ m∗ (B);
S
P∞
∗
(iii) m∗ ( ∞
n=1 An ) ≤
n=1 m (An ).
Suppose that A = {x1 , x2 , . . .} is a infinite countable subset of R; the finite case is
handled similarly.
−j
−j
Given ε > 0,
S put aj = xj − 2 ε and bj = xj + 2 ε for j = 1, 2, . . . .
Then A ⊂ ∞
j=1 (aj , bj ) and
∞
X
j=1
m∗ ((aj , bj )) = 2ε
Remark
∞
X
1
= 2ε.
2j
The second property is called monotonicity and the third
countable subadditivity.
j=1
Since ε was arbitrary, it follows that m∗ (A) = 0.
Corollary
The interval [0, 1] is uncountable.
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Properties of the Outer Lebesgue Measure
Measurable Sets
Definition (Carathéodory)
Proof.
A subset A of R is said to be measurable if
Only (iii) requires a proof.
Given ε > 0, suppose that An ⊂
∞
X
j=1
Then
(n)
(n)
S∞
(n) (n)
j=1 (aj , bj )
and
m∗ (B) = m∗ (B ∩ A) + m∗ (B ∩ Ac )
for any subset B of R. The collection of measurable subsets of R is
denoted M.
m∗ ((aj , bj )) ≤ m∗ (An ) + 2−n ε for n = 1, 2, . . . .
S∞
n=1 An
⊂
∞
∞ X
X
Remark
S∞ S∞
(n) (n)
j=1 (aj , bj )
n=1
(n) (n)
m∗ ((aj , bj ))
n=1 j=1
≤
∞
X
and
Notice that
m∗ (B) = m∗ ((B ∩ A) ∪ (B ∩ Ac )) ≤ m∗ (B ∩ A) + m∗ (B ∩ Ac ).
m∗ (An ) + ε.
Thus, a set A ⊂ R is measurable if and only if
n=1
S
P∞
∗
This means that m∗ ( ∞
n=1 An ) ≤
n=1 m (An ) + ε.
m∗ (B) ≥ m∗ (B ∩ A) + m∗ (B ∩ Ac )
Since ε was arbitrary, the statement follows.
for any subset B of R.
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Examples of Measurable Sets
Properties of Measurable Sets
Example
Definition
Clearly, R ∈ M and ∅ ∈ M.
A nonempty collection of sets C is called an algebra if
(i) A ∈ C implies that Ac ∈ C;
Proposition
(ii) A, B ∈ C implies that A ∪ B ∈ C.
Any subset of R with measure zero is measurable.
If, in addition,
Proof.
(iii) A1 , A2 , . . . ∈ C implies that
Suppose that A ⊂ R and m∗ (A) = 0.
If B ⊂ R, then B ∩ A ⊂ A, so m∗ (B ∩ A) ≤ m∗ (A) = 0.
Also, B ∩ Ac ⊂ B, so
∗
∗
c
∗
∗
then C is called a σ-algebra.
S∞
j=1 Aj
∈ C,
Theorem
c
m (B) ≥ m (B ∩ A ) = m (B ∩ A) + m (B ∩ A ).
The collection M is a σ-algebra.
Example
Remark
A consequence of the proposition is that any finite or countable subset of R
is measurable. For instance, Q is measurable.
The theorem is a consequence of the following series of
propositions.
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Properties of Measurable Sets
Properties of Measurable Sets
Proposition
Proposition
If A1 , A2 ∈ M, then A1 ∪ A2 ∈ M.
If A ∈ M, then Ac ∈ M.
Proof.
Suppose that B is a subset of R.
Since A2 ∈ M,
Proof.
m∗ (B ∩ Ac1 ) = m∗ (B ∩ Ac1 ∩ A2 ) + m∗ (B ∩ Ac1 ∩ Ac2 ).
This follows directly from the definition since B ∩ A = B ∩ (Ac )c
for any subset B of R.
Since B ∩ (A1 ∪ A2 ) = (B ∩ A1 ) ∪ (B ∩ Ac1 ∩ A2 ),
m∗ (B ∩ (A1 ∪ A2 )) ≤ m∗ (B ∩ A1 ) + m∗ (B ∩ Ac1 ∩ A2 ).
Remark
It follows that
De Morgan’s law shows that if A, B ∈ M, then
More generally,
T∞
m∗ (B ∩ (A1 ∪ A2 )) + m∗ (B ∩ (A1 ∪ A2 )c ) ≤ m∗ (B ∩ A1 ) + m∗ ((B ∩ Ac1 ) ∩ A2 )
+ m∗ ((B ∩ Ac1 ) ∩ Ac2 )
A ∩ B = (Ac ∪ B c )c ∈ M.
j=1 Aj
= m∗ (B ∩ A1 ) + m∗ (B ∩ Ac1 )
= m∗ (B),
∈ M if A1 , A2 , . . . ∈ M.
where the last equality is a consequence of the assumption that A1 ∈ M.
This shows that A1 ∪ A2 ∈ M.
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Properties of Measurable Sets
Properties of Measurable Sets
Proof.
Proposition (Finite additivity of the outer measure)
The statement is obviously true for n = 1. Suppose that it is true for some
n ≥ 1.
Because the sets A1 , . . . , An+1 are disjoint and An+1 is measurable, we then
have that
Suppose that A1 , . . . , An are measurable and pairwise disjoint
subsets of R and B is a subset of R. Then
n
n
X
[
m∗ B ∩
Aj =
m∗ (B ∩ Aj ).
j=1
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n+1
n+1
[ [
m∗ B ∩
Aj = m∗ (B ∩
Aj ) ∩ An+1
j=1
j=1
j=1
+ m∗ (B ∩
Remark
m
n
[
j=1
Aj =
n
X
j=1
Aj ) ∩ Acn+1
n
[
= m∗ (B ∩ An+1 ) + m∗ B ∩
Aj
Taking B = R, it follows that
∗
n+1
[
j=1
∗
m (Aj )
= m∗ (B ∩ An+1 ) +
j=1
n
X
j=1
m∗ (B ∩ Aj ).
This shows that the statement is true for n + 1.
if A1 , . . . , An are measurable and pairwise disjoint subsets of R.
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Properties of Measurable Sets
Properties of Measurable Sets
Proposition
If A1 , A2 , . . . ∈ M, then
Proof.
If A1 , A2 , . S
. . ∈ M, then
S∞there exist disjoint sets B1 , B2 , . . . ∈ M
such that ∞
B
=
j
j=1
j=1 Aj .
Put B1 = A1 and Bj = Aj \
Sj−1
k=1 Ak
j=1 Aj
∈ M.
According to the lemma, we can assume that A1 , A2 , . . . are disjoint.
Suppose that
SnR.
S B is a subset of
c
c
Put A = ∞
j=1 Aj and En =
j=1 Aj for n = 1, 2, . . . . Then En ∈ M and En ⊃ A for
every n, so
Lemma
Proof.
S∞
m∗ (B) = m∗ (B ∩ En ) + m∗ (B ∩ Enc ) ≥
n
X
j=1
m∗ (B ∩ Aj ) + m∗ (B ∩ Ac ).
The left-hand side of this inequality is independent of n, so it follows that
for j = 2, 3, . . . .
m∗ (B) ≥
∞
X
j=1
m∗ (B ∩ Aj ) + m∗ (B ∩ Ac ).
From this and the subadditivity of m∗ , it now follows that
m∗ (B) ≥ m∗ (B ∩ A) + m∗ (B ∩ Ac ),
which means that A is measurable.
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Properties of Measurable Sets
Properties of Measurable Sets
Proof: End
Lemma
It follows from the definition of the outer measure that
Any interval is measurable.
m∗ (B ∩ (a, ∞)) + m∗ (B ∩ (−∞, a]) ≤
Proof: Beginning
=
We first show that (a, ∞) is measurable for every a ∈ R.
∞
X
j=1
∞
X
j=1
Let B be a subset of R.
`(Ij0 ) +
∞
X
`(Ij00 )
j=1
`(Ij ) ≤ m∗ (B) + ε.
Since ε > 0 was arbitrary, we see that (a, ∞) is measurable.
This implies that the sets
Given ε > P
0, let (Ij )∞
j=1 be a covering of B with open intervals
∗
such that ∞
j=1 `(Ij ) ≤ m (B) + ε.
[a, ∞) =
Put Ij0 = Ij ∩ (a, ∞) and Ij00 = Ij ∩ (−∞, a] for j = 1, 2, . . . .
T∞
n=1 (a
− n1 , ∞)
(−∞, a] = (a, ∞)c and (−∞, a) = [a, ∞)c
00 ∞
Then (Ij0 )∞
j=1 and (Ij )j=1 are coverings of B ∩ (a, ∞) and
B ∩ (−∞, a], respectively.
(a, b) = (−∞, b) ∩ (a, ∞) and [a, b] = (−∞, b] ∩ [a, ∞)
(a, b] = (−∞, b] ∩ (a, ∞) and [a, b) = (−∞, b) ∩ [a, ∞)
are all measurable.
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Properties of Measurable Sets
The Lebesgue Measure
Proposition
Definition
Any open subset of R is measurable.
The Lebesgue measure m is the restriction of m∗ to M.
Proof.
Theorem
Any open subset of R is a countable union of open intervals.
The Lebesgue measure has the following properties:
Corollary
(i) m(I ) = `(I ) for any interval I ;
Any closed subset of R is measurable.
(ii) If A ∈ M, then A + x ∈ M and m(A + x) = m(A) for any
number x ∈ R;
(iii) If AS
M is a sequence of pairwise disjoint sets, then
1 , A2 , . .. ∈ P
∞
A
=
m ∞
j
j=1
j=1 m(Aj ).
Proof.
If F is a closed subset of R, then F =
F c is open and therefore measurable.
(F c )c
is measurable since
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The Lebesgue Measure
Proof.
The first property has already been proved and we leave the
second as an exercise.
S
Sn
Since ∞
j=1 Aj ⊃
j=1 Aj for any integer n ≥ 1, the finite
additivity of m∗ shows that
m
∞
[
j=1
n
n
[
X
Aj ≥ m
Aj =
m(Aj ).
j=1
Letting n → ∞, it follows that m
j=1
S∞
j=1 Aj
≥
P∞
j=1 m(Aj ).
The reverse inequality is a consequence of the subadditivity of
the outer measure m∗ .
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