Lebesgue Measure

Consider the Borel $\sigma$-field $\cB$ on the sample space $\Omega = (0, 1]$. We previously defined the collection $\cB_0$ as:

$$\cB_0 = \left\{ \text{finite unions of disjoint intervals of } \Omega \right\}$$

Recall from our discussion on $\sigma$-fields that $\cB_0$ is not a $\sigma$-field, but it is a field.

Motivation

Since the Borel $\sigma$-field $\cB$ is generated by $\cB_0$ (i.e., $\cB = \sigma(\cB_0)$), it is notoriously difficult to explicitly describe every element in $\cB$. In contrast, $\cB_0$ is very easy to describe—it just consists of unions of intervals!

This motivates our strategy for defining the Lebesgue measure:

1. Define a probability measure on the simple collection $\cB_0$.
2. Use the Extension Theorem to extend this measure to the complex collection $\cB$.

Defining the Measure on the Field

Let’s define a set function $\lambda_0$ on $\cB_0$ such that it assigns the “length” to every interval. For any interval $(a, b] \subseteq (0, 1]$, we define:

$$\lambda_0((a, b]) = b - a$$

For other members of $\cB_0$ (which are finite disjoint unions of intervals), $\lambda_0$ is defined as the sum of the lengths of the component intervals.

It is straightforward to verify that $\lambda_0$ satisfies the probability axioms on the field $\cB_0$.

The Lebesgue Measure

We now invoke the Extension Result.

Extension to Borel σ-field

Recall that the Borel $\sigma$-field is $\cB = \sigma(\cB_0)$. Hence, by the Extension Theorem and the Uniqueness Theorem, the pre-measure $\lambda_0$ defined on $\cB_0$ uniquely extends to a probability measure on $\cB$.

We denote this extension by $\lambda$ and call it the Lebesgue measure on $\cB$.

In other words, the Lebesgue measure is the only probability measure on the measurable space $((0, 1], \cB)$ that assigns the length $b-a$ to every interval $(a, b]$.

Summary

This process turns our measurable space into a probability space (or generally, a measure space):

$$\underbrace{((0, 1], \cB)}_{\text{Measurable Space}} \xrightarrow{\text{add } \lambda} \underbrace{((0, 1], \cB, \lambda)}_{\text{Measure Space}}$$

Measure Space vs. Probability Space

The resulting triplet $((0, 1], \cB, \lambda)$ is a measure space. Since $\lambda((0, 1]) = 1 - 0 = 1$, it is specifically a probability space (a measure space with total mass equal to 1).