$
\def\P{\mathsf{\sf P}}
\def\E{\mathsf{\sf E}}
\def\Var{\mathsf{\sf Var}}
\def\Cov{\mathsf{\sf Cov}}
\def\std{\mathsf{\sf std}}
\def\Cor{\mathsf{\sf Cor}}
\def\R{\mathbb{R}}
\def\c{\,|\,}
\def\bb{\boldsymbol}
\def\diag{\mathsf{\sf diag}}
\def\defeq{\stackrel{\tiny\text{def}}{=}}
$

We discuss three important measure functions: the discrete measure, the Lebesgue measure, and the Lebesgue-Stieltjes measure.

In this section, we consider a general method to define measures on a finite $\Omega$ or a countably infinite $\Omega$. In this case, we use the $\sigma$-algebra $2^{\Omega}$ - the power set of $\Omega$.

We define the discrete measure on the measurable space $(\Omega,2^{\Omega})$ associated with a set of non-negative numbers $\{p_{\omega}: \omega\in\Omega\}$ as \[\mu(A)=\sum_{\omega\in A} p_{\omega}.\] The function $\mu:2^{\Omega} \to \R$ satisfies $\mu(\emptyset)=0$ and is countably additive, implying that it is a measure. If $\sum_{\omega\in\Omega} p_{\omega}=1$, the discrete measure $\mu$ is also a probability measure $\P$.

We start by defining the $\sigma$-algebra that will be used to define the Lebesgue measure.

Proposition E.5.1 and Proposition E.1.5 imply that $\mathcal{B}((0,1])$ is generated by the open balls in $(0,1]$, or intervals $(a,b)$ with $0\leq a < b\leq 1$. To show that $\mathcal{B}((0,1])=\sigma(\mathcal{D})$ it suffices to show that (i) $(a,b)\in \sigma(\mathcal{D})$, and that (ii) $(a,b]\in \mathcal{B}((0,1])$. The first claim holds since $(a,b)=\cup_{n\in\mathbb{N}} (a,b-1/n]$ and the second holds since $(a,b]=\cap_{n\in\mathbb{N}} (a,b+1/n) = (\cup_{n\in\mathbb{N}} (a,b+1/n)^c)^c$.

Since $\mathcal{D}$ is an algebra and $\sigma(\mathcal{D})=\mathcal{B}(0,1])$ (by the proposition above), we can use Caratheodory's extension theorem (Proposition E.3.10) to assert that there is a unique extension of $\mu$ to the Borel $\sigma$-algebra $\mathcal{B}((\alpha,\beta])$.

We make the following observations.

- The Lebesgue measure may be defined on different types of intervals $(\alpha,\beta)$, $[\alpha,\beta]$, or $[\alpha,\beta)$ in accordance with the definition above on intervals of the form $[\alpha,\beta)$. Consistency of the Lebesgue measure over all of these intervals follows from the fact that the Lebesgue measure (length) of a single point is zero.
- When $\beta-\alpha=1$, the Lebesgue measure is also a probability measure. In other cases, we can define a probability measure associated with the Lebesgue measure on $(\alpha,\beta]$ as follows \[ \P(A) = \frac{1}{\beta-\alpha}\, \mu(A).\] This probability function is the classical probability function on continuous spaces, defined in Section 1.4.
- Letting $\alpha\to-\infty$ and $\beta\to+\infty$ generalizes the Lebesgue measure to $\Omega=\R$ and the $\sigma$-algebra $\mathcal{B}(\R)$. Note that in this case $\mu(\Omega)=+\infty$, implying that the Lebesgue measure on $\R$ is not a probability measure, and cannot be converted to one using the normalization that is described above.
- The Lebesgue measure can be generalized to subsets of higher dimensional Euclidean spaces $\R^d$. The resulting measure generalizes area and volume for the cases $d=2$ and $d=3$, respectively. These generalizations will be developed in Section F.5.

As in the case of the Lebesgue measure, letting $\alpha\to-\infty$ and $\beta\to\infty$ generalizes the Lebesgue-Stieltjes measure to $\Omega=\R$ and the $\sigma$-algebra $\mathcal{B}(\R)$.