Lebesgue Measurable Sets Review

# Lebesgue Measurable Sets Review

We will now review some of the recent material regarding Lebesgue measurable sets.

- Recall from the
**Lebesgue Measurable Sets**page that a set $E \in \mathcal P (\mathbb{R})$ is said to be**Lebesgue Measurable**if for all $A \in \mathcal P(\mathbb{R})$ we have that:

\begin{align} \quad m^*(A) = m^*(A \cap E) + m^*(A \cap E^c) \end{align}

- We noted that $m^*(A) \leq m^*(A \cap E) + m^*(A \cap E^c)$ always holds, and so, to show that $E$ is Lebesgue measurable we need to only prove that $m^*(A) \geq m^*(A \cap E) + m^*(A \cap E^c)$.

- We then proved some elementary results for Lebesgue measurable sets which are summarized below.

Proposition 1: |
If $E \in \mathcal P (\mathbb{R})$ then $E$ is Lebesgue measurable if and only if $E^c$ is Lebesgue measurable. |

Proposition 2: |
The sets $\emptyset$ and $\mathbb{R}$ are Lebesgue measurable. |

Proposition 3: |
If $E \in \mathcal P (\mathbb{R})$ and $m^*(E) = 0$ then $E$ is Lebesgue measurable. |

- We then proved
**A Property of Finite Mutually Disjoint Collections of Lebesgue Measurable Sets**by mathematical induction. This property said that if $A \in \mathcal P (\mathbb{R})$ and $(E_k)_{k=1}^{n}$ is a finite sequence of mutually disjoint Lebesgue measurable sets then:

\begin{align} \quad m^* \left ( A \cap \bigcup_{k=1}^{n} E_k \right ) = \sum_{k=1}^{n} m^*(A \cap E_k) \end{align}

- On
**The Union of a Finite Collection of Lebesgue Measurable Sets is Lebesgue Measurable**page we proved that if $E_1$ and $E_2$ are Lebesgue measurable then $E_1 \cup E_2$ is Lebesgue measurable. As a consequence we saw that the set $\mathcal M$ of Lebesgue measurable sets is an algebra.

- On
**The Union of a Countable Collection of Lebesgue Measurable Sets is Lebesgue Measurable**page we further proved that if $(E_n)_{n=1}^{\infty}$ is a sequence of Lebesgue measurable sets then $\displaystyle{\bigcup_{n=1}^{\infty} E_n}$ is a Lebesgue measurable set. As a consequence we saw that the set $\mathcal M$ of Lebesgue measurable sets is a $\sigma$-algebra.

- On
**The Lebesgue Measurability of Intervals**page we began classifying what sets were Lebesgue measurable. We proved that every interval $I \in \mathcal P (\mathbb{R})$ is Lebesgue measurable.

- On
**The Lebesgue Measurability of Translates of Lebesgue Measurable Sets**page we proved that if $E \in \mathcal P (\mathbb{R})$ is Lebesgue measurable then for all $a \in \mathbb{R}$, the translate $E + a = \{ e + a : e \in E \}$ is Lebesgue measurable.

- On
**The Regularity Properties of the Lebesgue Measure**we looked at some equivalent statements for a subset of $E$ of $\mathbb{R}$ to be Lebesgue measurable. We restate that result below:

Theorem (The Regularity Properties of the Lebesgue Measure): Let $E \in \mathcal P(\mathbb{R})$. Then the following statements are equivalent:a) $E$ is Lebesgue measurable.b) For all $\epsilon > 0$ there exists an open set $O$ with $E \subseteq O$ such that $m^*(O \setminus E) < \epsilon$.c) There exists a $G_{\delta}$ set $G$ such that $E \subseteq G$ and $m^*(G \setminus E) = 0$.d) For all $\epsilon > 0$ there exists a closed set $F$ with $F \subseteq E$ such that $m^*(E \setminus F) < \epsilon$.e) There exists an $F_{\sigma}$ set $F$ such that $F \subseteq E$ and $m^*(E \setminus F) = 0$. |

- On
**The Collection of Lebesgue Measurable Sets**page we noted that the set $\mathcal M$ of Lebesgue measurable sets also contains every $G_{\delta}$-set (countable intersections of open sets) and every $F_{\sigma}$-set (countable unions of closed sets). We also saw that $\mathcal M$ contains the $\sigma$-algebra of Borel sets (the smallest $\sigma$-algebra containing the collection of open sets in $\mathbb{R}$).