Lebesgue Measure

Outer Measure is Additive when One Set is Open

Suppose that

A, G \subseteq ℝ

A \cap G = \emptyset

, and

G

is open. Then

| A \cup G | = | A | + | G | .

Outer Measure is Additive when One Set is Closed

Suppose that

A, F \subseteq ℝ

A \cap F = \emptyset

, and

F

is closed. Then

| A \cup F | = | A | + | F | .

Borel Sets Can be Approximated from Below by Closed Sets

Suppose that

B \subseteq ℝ

is a Borel set. For every

ϵ \in ℝ^{+}

, there exists a closed set

F \subseteq B

such that

| B ∖ F | < ϵ .

Outer Measure is Additive when One Set is Borel

Suppose that

A, B \subseteq ℝ

A \cap B = \emptyset

, and

B

is a Borel set. Then

| A \cup B | = | A | + | B | .

There is a Non-Borel Set of Finite Outer Measure

There exists a set

B \subseteq ℝ

such that

| B | < \infty

and

B

is not a Borel set.

Outer Measure is a Measure on Borel Sets

Let

ℬ

be the Borel

σ

-algebra on

ℝ

. Outer measure restricted to

ℬ

is a measure on

(ℝ, ℬ)

Lebesgue Measure on Borel Sets

Let

ℬ

be the Borel

σ

-algebra on

ℝ

. Lebesgue measure on Borel sets is the measure on

(ℝ, ℬ)

that sends each Borel set to its outer measure.

Lebesgue Measurable Set

A set

A \subseteq ℝ

is Lebesgue measurable if there exists a Borel set

B \subseteq A

such that

| A ∖ B | = 0.

Equivalent Characterizations of Lebesgue Measurable Sets

Suppose that

A \subseteq ℝ

. The following are equivalent:

$A$ is Lebesgue measurable.
For every $ϵ \in ℝ^{+}$ , there exists a closed set $F \subseteq A$ such that $| A ∖ F | < ϵ$ .
There are closed sets $F_{1}, F_{2}, \dots \subseteq A$ such that $| A ∖ ⋃_{k = 1}^{\infty} F_{k} | = 0$ .
There exists a Borel set $B \subseteq A$ such that $| A ∖ B | = 0$ .
For every $ϵ \in ℝ^{+}$ , there exists an open set $G \supseteq A$ such that $| G ∖ A | < ϵ$ .
There are open sets $G_{1}, G_{2}, \dots \supseteq A$ such that $| (⋂_{k = 1}^{\infty} G_{k}) ∖ A | = 0$ .
There exists a Borel set $B \supseteq A$ such that $| B ∖ A | = 0$ .

Outer Measure is a Measure on Lebesgue Measurable Sets

The Lebesgue measurable subsets of

ℝ

form a

σ

-algebra

ℒ

, and outer measure restricted to

ℒ

is a measure on

(ℝ, ℒ)

Lebesgue Measure on Lebesgue Measurable Sets

Let

ℒ

be the

σ

-algebra of Lebesgue measurable subsets of

ℝ

. Lebesgue measure on Lebesgue measurable sets is the measure on

(ℝ, ℒ)

that sends each Lebesgue measurable set to its outer measure.

Cantor Set

Define

G_{1} : = (1 / 3, 2 / 3)

. For

n > 1

, let

G_{n}

be the union of the open middle thirds of the intervals remaining in

[0, 1] ∖ ⋃_{j = 1}^{n - 1} G_{j} .

The Cantor set is

C : = [0, 1] ∖ ⋃_{n = 1}^{\infty} G_{n} .

Base Three Description of the Cantor Set

The Cantor set is the set of all

x \in [0, 1]

that have a base

3

representation using only the digits

0

and

2

Basic Properties of the Cantor Set

The Cantor set

C

is closed, has Lebesgue measure

0

, and contains no interval with more than one element.

Cantor Function

The Cantor function

Λ : [0, 1] \to [0, 1]

is defined by converting base

3

representations to base

2

representations: for

x \in C

, replace each digit

2

in the

0

-and-

2

base

3

representation by

1

; for

x \notin C

, truncate its base

3

representation immediately after the first digit

1

, replace any earlier

2

's by

1

's, and read the result as a base

2

number.

Cantor Function

The Cantor function

Λ

is a continuous increasing function from

[0, 1]

onto

[0, 1]

. Moreover,

Λ (C) = [0, 1] .

Cantor Set is Uncountable

The Cantor set is uncountable.

Continuous Image of a Lebesgue Measurable Set Can be Nonmeasurable

There exists a Lebesgue measurable set

A \subseteq [0, 1]

such that

| A | = 0

and

Λ (A)

is not Lebesgue measurable.