Outer Measure

Length of an Open Interval

The length

ℓ (I)

of an open interval

I \subseteq ℝ

is defined by

ℓ (I) = {\begin{matrix} b - a & a m p; I = (a, b) with a < b, \\ 0 & a m p; I = \emptyset, \\ \infty & a m p; I = (- \infty, a) or I = (a, \infty), \\ \infty & a m p; I = (- \infty, \infty) . \end{matrix}

Outer Measure

Suppose that

A \subseteq ℝ

. The outer measure of

A

| A | : = \inf {\sum_{k = 1}^{\infty} ℓ (I_{k}) : A \subseteq ⋃_{k = 1}^{\infty} I_{k} and each I_{k} is an open interval} .

Countable Sets Have Outer Measure Zero

Every countable subset of

ℝ

has outer measure

0

Outer Measure Preserves Order

Suppose that

A, B \subseteq ℝ

and

A \subseteq B

. Then

| A | \leq | B | .

Translation of a Set

Suppose that

t \in ℝ

and

A \subseteq ℝ

. Define the translation of

A

t

t + A : = {t + a : a \in A} .

Outer Measure is Translation Invariant

Suppose that

t \in ℝ

and

A \subseteq ℝ

. Then

| t + A | = | A | .

Countable Subadditivity of Outer Measure

Suppose that

A_{1}, A_{2}, \dots \subseteq ℝ

. Then

| ⋃_{k = 1}^{\infty} A_{k} | \leq \sum_{k = 1}^{\infty} | A_{k} | .

Open Cover

Suppose that

A \subseteq ℝ

. A collection

𝒞

of open subsets of

ℝ

is an open cover of

A

A \subseteq ⋃_{G \in 𝒞} G .

Finite Subcover

Suppose that

𝒞

is an open cover of

A \subseteq ℝ

. We say that

𝒞

has a finite subcover if there are

G_{1}, \dots, G_{n} \in 𝒞

such that

A \subseteq G_{1} \cup \dots \cup G_{n} .

Heine-Borel Theorem

Every open cover of a closed bounded subset of

ℝ

has a finite subcover.

Outer Measure of a Closed Interval

Suppose that

a, b \in ℝ

and

a < b

. Then

| [a, b] | = b - a .

Nontrivial Intervals are Uncountable

Every interval in

ℝ

that contains at least two distinct elements is uncountable.

Outer Measure is Not Additive

There exist disjoint subsets

A, B \subseteq ℝ

such that

| A \cup B | \neq | A | + | B | .

No Countably Additive Translation Invariant Extension of Length to All Subsets

There is no function

μ : 𝒫 (ℝ) \to [0, \infty]

such that

$μ (I) = ℓ (I)$ for every open interval $I \subseteq ℝ$ ,
$μ (⋃_{k = 1}^{\infty} A_{k}) = \sum_{k = 1}^{\infty} μ (A_{k})$ for every pairwise disjoint sequence $A_{1}, A_{2}, \dots \subseteq ℝ$ ,
$μ (t + A) = μ (A)$ for every $t \in ℝ$ and every $A \subseteq ℝ$ .