> [!definition] > > Let $\mu$ be a [[Measure Space|measure]] on a [[Topological Space|topological space]] $(X, \topo)$. $\mu$ is a **Borel** measure if $\mu: \cb_X \to \real$ its domain is the [[Borel Sigma Algebra|Borel sigma-algebra]] on $X$. > [!definition] > > Let $-\infty \le a < b < \infty$, then the sets $\emptyset$, $(a, b]$, and $(a, \infty)$ are known as **h-intervals** (half-open). The h-intervals form an [[Elementary Family|elementary family]], and the collection $\alg$ of disjoint unions of h-intervals is an [[Algebra|algebra]]. > > *Proof*. Let $-\infty \le a < b < \infty$, and $-\infty \le c < d < \infty$ then > $ > \begin{align*} > (a, b] \cap (c, d] &= (\max(a, c), \min(b, d)] \\ > (a, b] \cap (c, \infty) &= (\max(a, c), b] \\ > (a, \infty) \cap (c, \infty) &= (\max(a, c), \infty) > \end{align*} > $ > the h-intervals are closed under finite intersections. Moreover, > $ > \begin{align*} > (a, b]^c &= (-\infty, a] \cup (b, \infty) \\ > (a, \infty)^c &= (-\infty, a] \\ > \emptyset^c &= (-\infty, \infty) > \end{align*} > $ > the complement of any h-interval can be expressed as a disjoint union of up to two h-intervals. > [!definition] > > Let $\mu: \cb_\real \to \real$ be a finite Borel measure on $\real$, and let $F(x) = \mu\paren{(-\infty, x]}$. Then $F$ is the **distribution function** of $\mu$ and > - $F$ is increasing. > - $F$ is right [[Continuity|continuous]]. > - For any $b > a$, $\mu((a, b]) = F(b) - F(a)$. > > *Proof*. Let $b > a \in \real$, then > $ > (-\infty, a] \subset (-\infty, b] \Rightarrow \mu((-\infty, a]) \le \mu((-\infty, b]) > $ > Let $\bracs{x_i}_1^\infty$ be an decreasing sequence with limit $x$, then > $ > \limv{i}\mu((-\infty, x_i]) = \mu\paren{\bigcap_{i \in \nat}(-\infty, x_i]} = \mu((-\infty, x]) > $ > by continuity from above. Lastly, > $ > \begin{align*} > (-\infty, a] \cup (a, b] &= (-\infty, b] \\ > \mu((-\infty, a]) + \mu((a, b])&= \mu((-\infty, b]) \\ > \mu((a, b]) &= \mu((-\infty, b]) - \mu((-\infty, a]) > \end{align*} > $