The Lebesgue Outer Measure Review

# 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:

\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:

\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:

\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:

\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:

\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:

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