The Lebesgue Outer Measure Review

We will now review some of the recent material regarding the Lebesgue outer measure.

• Recall from The Length Function on the Set of Intervals page that if $I \in \mathcal P(\mathbb{R})$ is an interval with endpoints $a, b \in \mathbb{R}^*$, $a < b$ then the Length of $I$ is defined as:

(1)

\begin{align} \quad l(I) = \left\{\begin{matrix}b - a & \mathrm{if} \: I \: \mathrm{is \: bounded} \\ \infty & \mathrm{if} \: I \: \mathrm{is \: unbounded} \end{matrix}\right. \end{align}

• On The Lebesgue Outer Measure page we said that if $E \in \mathcal P(\mathbb{R})$ then the Lebesgue Outer Measure of $E$ is defined as:

(2)

\begin{align} \quad m^*(E) = \inf \left \{ \sum_{n=1}^{\infty} l(I_n) : E \subseteq \bigcup_{n=1}^{\infty} I_n \: \mathrm{and} \: \{ I_n = (a_n, b_n) \}_{n=1}^{\infty} \right \} \end{align}

• In other words, the Lebesgue outer measure of $E$ is the infimum of the sum of the lengths of all open interval covers of $E$.

• We then proved an important result known as the Monotonicity Property of the Lebesgue Outer Measure. This property says that if $A, B \in \mathcal P (\mathbb{R})$ then:

(3)

\begin{align} \quad m^*(A) \leq m^*(B) \end{align}

• On The Lebesgue Outer Measure of Intervals page we proved that if $I \in \mathcal P(\mathbb{R})$ is an interval then:

(4)

\begin{align} \quad m^*(I) = l(I) \end{align}

• On the Countable Subadditivity of the Lebesgue Outer Measure page we looked at another important property known as the Countable Subadditivity of the Lebesgue Outer Measure. This property says that if $(A_n)_{n=1}^{\infty}$ is any sequence of subsets of $\mathbb{R}$ then the Lebesgue outer measure of the union of the $A_n$s is less than or equal to the sum of the Lebesgue outer measures of the $A_n$s, that is:

(5)

\begin{align} \quad m^* \left ( \bigcup_{n=1}^{\infty} A_n \right ) \leq \sum_{n=1}^{\infty} m^* (A_n) \end{align}

• As a nice corollary, we saw that if $E \in \mathcal P(\mathbb{R})$ is a countable set then $m^*(E) = 0$.

• On the Translation Invariance of the Lebesgue Outer Measure page we looked at a third important property known as the Translation Invariance of the Lebesgue Outer Measure. This property says if $E \in \mathcal P(\mathbb{R})$ and $a \in \mathbb{R}$ with $E + a = \{ e + a : e \in E \}$ then:

(6)

\begin{align} \quad m^*(E + a) = m^*(E) \end{align}

• Lastly, on The Lebesgue Outer Measure of Open Sets and G𝛿-Sets Containing a Set page we proved two important results regarding the Lebesgue outer measure which are summarized in the table below.

Property 1:	If $A \in \mathcal P(\mathbb{R})$ then for all $\epsilon > 0$ there exists an open set $O$ with $A \subseteq O$ such that $m^*(O) \leq m^*(A) + \epsilon$.
Property 2:	If $A \in \mathcal P(\mathbb{R})$ then there exists a $G_{\delta}$-set $G$ with $A \subseteq G$ such that $m^*(A) = m^*(G)$.