The Lebesgue Measure

We have already defined the Lebesgue outer measure, $m^*$, which is a set function defined for all subsets of $\mathbb{R}$. We said that if $E \subseteq \mathbb{R}$ then the Lebesgue outer measure of $E$ is defined as:

(1)

\begin{align} \quad m^*(E) = \inf \left \{ \sum_{n=1}^{\infty} l(I_k) : E \subseteq \bigcup_{n=1}^{\infty} I_n, \: \{ I_n = (a_n, b_n) \}_{n=1}^{\infty} \right \} \end{align}

We have already proven that the set of Lebesgue measurable sets $\mathcal M$ is a $\sigma$-algebra.

As a result, we can now define the Lebesgue measure. This set function which we will denote by $m$ will be defined on the set of Lebesgue measurable functions and is in fact equal to the Lebesgue outer measure. In other words, the Lebesgue measure will be the set function that is the Lebesgue outer measure restricted to $\mathcal M$.

Definition: The Lebesgue Measure is the set function $m : \mathcal M \to [0, \infty) \cup \{ \infty \}$ defined for all Lebesgue measurable sets $E$ by $m(E) = m^*(E)$.

It is important to note that we have not really introduced anything new here. However, we will see that restriction, $m$, will have additional properties that $m^*$ does not have due to limiting our domain to Lebesgue measurable sets.