The Lebesgue Measure Review

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

• On The Lebesgue Measure page we defined the Lebesgue Measure to be the set function $m : \mathcal M \to [0, \infty) \cup \{ \infty \}$ for all Lebesgue measurable sets $E$ by $m(E) = m^*(E)$. In other words, the Lebesgue measure is the Lebesgue outer measure set function restricted to the set of Lebesgue measurable functions.

• On The Excision Property of the Lebesgue Measure page we proved the important excision property of the Lebesgue measure which states that if $A$ and $B$ are Lebesgue measurable sets with $A \subseteq B$ and $m(A) < \infty$ then:

(1)

\begin{align} \quad m(B \setminus A) = m(B) - m(A) \end{align}

• On The Lebesgue Measure of a Countable Union of Mutually Disjoint Lebesgue Measurable Sets page we proved that if $(E_n)_{n=1}^{\infty}$ is a countable sequence of disjoint Lebesgue measurable sets then:

(2)

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

• On The Continuity Properties of the Lebesgue Measure page we looked at two important continuity properties of the Lebesgue measure. First, if $\{ A_n \}_{n=1}^{\infty}$ is a collection of Lebesgue measurable sets such that $A_n \subseteq A_{n+1}$ for all $n \in \mathbb{N}$ then:

(3)

\begin{align} \quad m \left ( \bigcup_{n=1}^{\infty} A_n \right ) = \lim_{n \to \infty} m(A_n) \end{align}

• Additionally, if $\{ B_n \}_{n=1}^{\infty}$ is a collection of Lebesgue measurable sets such that $B_n \supseteq B_{n+1}$ for all $n \in \mathbb{N}$ and if $m(B_1) < \infty$ then:

(4)

\begin{align} \quad m \left ( \bigcap_{n=1}^{\infty} B_n \right ) = \lim_{n \to \infty} m(B_n) \end{align}

• On The Cantor Set page we then defined the Cantor Set $C$ and proved that $C$ is an uncountable Lebesgue measurable set with $m(C) = 0$.