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:
(1)
\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.
(2)
\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}
 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 $\mathbb{R}$).