> [!definition] > > Let $(X, \topo)$ be a [[Topological Space|topological space]], then the [[Sigma Algebra|sigma algebra]] generated by the [[Open Set|open sets]] $\cb_X = \agb{\topo}$ is called the **Borel $\sigma$-algebra** on $X$, whose members are called **Borel sets**. > > $\cb_X$ includes open and [[Closed Set|closed]] sets along with their countable unions and intersections, and much more. > [!theorem] Some Notations > > | Description | Notation | > | ---- | ---- | > | Countable Intersection... | subscript $\delta$ | > | Countable Union... | subscript $\sigma$ | > | ...of open sets | $G$ | > | ...of closed sets | $F$ | > > Examples: $G_\delta$ is a countable union of open sets. $F_\sigma$ is a countable intersection of closed sets. $G_{\delta\sigma}$ is a countable intersection of $G_\delta$ sets. > [!theorem] > > $\cb_\real$ is generated by each of the following: > 1. The open intervals: $\ce_1 = \bracs{(a, b): a < b}$. > 2. The closed intervals: $\ce_2 = \bracs{[a, b]: a < b}$. > 3. The half-open intervals: $\ce_3 = \bracs{(a, b]: a < b}$, $\ce_4 = \bracs{[a, b): a < b}$. > 4. The open rays: $\ce_5= \bracs{(a, \infty): a \in \real}$, $\ce_6 = \bracs{(-\infty, a): a \in \real}$. > 5. The closed rays: $\ce_7 = \bracs{[a, \infty): a \in \real}$, $\ce_8 = \bracs{(-\infty, a]}$. > > *Proof*. We know that every [[Open Set|open set]] in $\real$ is a countable union of open intervals. Therefore $\ce_1$ generates $\cb_\real$. > > The closed intervals and half-open intervals can generate open intervals through countable intersections (or just straight up complements). > > The open rays can generate open intervals through complement intersections. > > The closed rays can generate closed intervals through complement intersections. > [!theorem] > > Let $X_1, \cdots, X_n$ be [[Metric Space|metric spaces]] and let $X = \prod_{j = 1}^{n}X_j$ be their [[Cartesian Product|Cartesian product]], equipped with the product metric. Then $\bigotimes_{j = 1}^{n}\cb_{X_j} \subseteq \cb_X$. If the $X_j$s are [[Separable Metric Space|separable]], then $\bigotimes_{j = 1}^{n}\cb_{X_j} = \cb_X$. > > *Proof*. $\bigotimes_{j = 1}^{n}\cb_{X_j}$ is generated by the sets $\pi^{-1}_j(U_j)$, where each $U_j$ is [[Open Set|open]] in $X_j$. Since these sets are open in $X$, $\bigotimes_{j = 1}^{n}\cb_{X_j} \subseteq \cb_X$. > > Suppose now that $C_j$ is a countable [[Dense|dense]] set in $X_j$, and let $\ce_j$ be the collection of open balls in $X_j$ with rational radius and centre in $C_j$. Then every open set in $X_j$ is a union of members of $\ce_j$. Moreover, the set of points in $X$ whose $j$-th coordinate is in $C_j$ for all $j$ is a countable dense subset of $X$, and balls of radius $r$ in $X$ can be produced by balls of radius $r$ in the $X_j$s. Therefore $\cb_X = \bigotimes_{j = 1}^{n}\cb_{X_j}$. > [!theorem] > > $ > \cb_{\real^n} = \bigotimes_{j = 1}^{n}\cb_{\real} > $ > > *Proof.* By previous theorem. > [!definition] > > Let $\ol{\real} = [-\infty, \infty]$ be the extended real numbers. Let > $ > \cb_{\ol{\real}} = \bracs{E \subseteq \ol\real: E \cap \real \in \cb_\real} > $ > be the Borel $\sigma$-algebra on $\ol{\real}$. > [!theorem] > > Let $\cb_{\real}$ be the Borel $\sigma$-algebra on $\ol\real$. Then $\cb_\real$ is generated by the rays $(a, \infty]$ or $[-\infty, a)$, $a \in \real$. > > *Proof*. We isolate the infinities > $ > \begin{align*} > \bracs{\infty} = \bigcap_{i \in \nat}(i, \infty] > &\quad > \bracs{-\infty} = \bigcap_{i \in \nat}[-\infty, -i) \\ > \bracs{\infty} = \bigcap_{i \in \nat}[i, \infty] > &\quad > \bracs{-\infty} = \bigcap_{i \in \nat}[-\infty, -i] \\ > \end{align*} > $ > and collect the h-intervals > $ > \begin{align*} > (a, b] &= (a, \infty] \cap ((b, \infty]^c)\\ > [a, b) &= [-\infty, b) \cap ([-\infty, a)^c) > \end{align*} > $ > generate $\cb_\real$, and attach the infinities to them to obtain $\cb_{\ol\real}$.