Example Problems Regarding Lebesgue Measurable Sets 1

So far we have defined a subset $E$ of $\mathbb{R}$ to be a Lebesgue measurable set if for all subsets $A$ of $\mathbb{R}$ we have that:

(1)

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

We noted that to show that a set of Lebesgue measurable it suffices to show that:

(2)

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

We have already proven the following results regarding Lebesgue measurable sets.

• 1. If $E$ is Lebesgue measurable then $E^c$ is Lebesgue measurable. Furthermore, $\emptyset$ and $\mathbb{R}$ are Lebesgue measurable.

• 2. If $m^*(E) = 0$ then $E$ is a Lebesgue measurable set.

• 3. If $A$ is any set and $\{ E_k \}_{k=1}^{n}$ is a finite collection of disjoint Lebesgue measurable sets then:

(3)

\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}

(4)

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

• 4. The union of a finite collection of Lebesgue measurable sets is Lebesgue measurable.

• 5. The union of a countable collection of Lebesgue measurable sets is Lebesgue measurable.

• 6. Every interval $I$ is Lebesgue measurable.

• 7. If $E$ is a Lebesgue measurable set then for all $a \in \mathbb{R}$ the translate $E + a$ is a Lebesgue measurable set.

• 8. 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 $m^*(E \setminus F) = 0$.

We will now look at some example problems regarding Lebesgue measurable sets. More problems can be found on the Example Problems Regarding Lebesgue Measurable Sets 2 page.

Example 1

Let $E \subseteq \mathbb{R}$ be such that $m^*(E) > 0$. Prove that there exists a subset $E' \subseteq E$ that is bounded such that $m^*(E') > 0$.

Suppose not. In other words, suppose that for every bounded subset $E' \subseteq E$ we have that $m(E') = 0$.

For each $k \in \mathbb{Z}$ let:

(5)

\begin{align} \quad I_k = [k, k+1] \end{align}

Then $(I_k)_{k \in \mathbb{Z}}$ is a countable collection of closed and bounded intervals. For each $k \in \mathbb{Z}$ let:

(6)

\begin{align} \quad E_k = E \cap I_k \end{align}

Then $(E_k)_{k \in \mathbb{Z}}$ is a countable collection of sets whose union is all of $E$. Furthermore, for each $k \in \mathbb{Z}$ we have that $E_k$ is bounded and $E_k \subseteq E$. Therefore by hypothesis, for each $k \in \mathbb{Z}$:

(7)

\begin{align} \quad m^*(E_k) = 0 \end{align}

Now from the countable subadditivity of the Lebesgue outer measure we have that:

(8)

\begin{align} \quad m^*(E) = m^* \left ( \bigcup_{k \in \mathbb{Z}} E_k \right ) \leq \sum_{k \in \mathbb{Z}} m^*(E_k) = 0 \end{align}

This contradicts $m^*(E) > 0$. So there must exists a bounded subset $E' \subseteq E$ such that $m^*(E') > 0$.

Example 2

Let $E \subseteq \mathbb{R}$ be such that $m(E) = 0$. Prove that for all $a \in \mathbb{R}$, we have that $m(E + a) = 0$.

Since $m(E) = 0$ we have that $E$ is a Lebesgue measurable set. We know that the translate of every Lebesgue measurable set is Lebesgue measurable, so $E + a$ is Lebesgue measurable, and furthermore:

(9)

\begin{align} \quad m(E + a) = m(E) = 0 \end{align}

Example 3

Prove that every open set in $\mathbb{R}$ is Lebesgue measurable. Prove that every closed in $\mathbb{R}$ is Lebesgue measurable.

Let $O \subseteq \mathbb{R}$ be an open set. We know that every open set in $\mathbb{R}$ is a countable disjoint union of open intervals, that is, $\displaystyle{O = \bigcup_{n=1}^{\infty} I_n}$ where each $I_n$ is an open interval. We have already seen that every open interval is Lebesgue measurable. So $O$ is a countable union of Lebesgue measurable sets, so $O$ is Lebesgue measurable.

Let $C \subseteq \mathbb{R}$ be a closed set. Then $C^c$ is an open set. So $C^c$ is a Lebesgue measurable set. But the complement of a Lebesgue measurable set is Lebesgue measurable. So $C = (C^c)^c$ is Lebesgue measurable.