My mathematical statistics book denotes $\sigma$-field as following:
Let $\Bbb B$ be the collection of subsets of $\Bbb C$ where $\Bbb C$ denotes sample space which is the collection of all possible events. Then $\Bbb B$ is $\sigma$-field if
(1) $\emptyset \in \Bbb B$ and $\exists b \in \Bbb B$ s.t. $\emptyset \subset b$
(2) $C \in \Bbb B \Rightarrow C^c\in \Bbb B $ where $C \in \Bbb C$
(3) $\{C_1, C_2, C_3..\} \in \Bbb B \Rightarrow \cup_{i=1}^{\infty}C_i \in \Bbb B$ where $\{C_1, C_2, C_3..\}$ is countable collection of subsets of $\Bbb C$
Is this field a specific example of Borel Field? or this field is eqaully defined with Borel Field?
$\endgroup$ 23 Answers
$\begingroup$Borel field is a special case of Sigma field. It is a Sigma algebra generated by a collection of subsets of C (or omega in most sources) whose elements are "finite open intervals on Real numbers").
The definition you wrote defines Sigma field in general. So to make it into Borel field, you need to define it only on a set A of all finite open intervals on R. It will give you the smallest Sigma field containing A.
Source: Mathematical Statistics by Jun Shao
$\endgroup$ $\begingroup$A sigma field on a non-emptyset $X$ is a collection $\mathcal{F}\subseteq 2^X$ that contains $\emptyset$, is closed under complementation and is closed under countable unions.
A Borel field is a sigma field $\mathcal{F}$ that is defined on a topological space $(X, \mathcal{T})$ such that $\mathcal{T} \subseteq \mathcal{F}$. Most authors require that $\mathcal{F}$ is generated by the topology $\mathcal{T}$, that is $\mathcal{F}$ is the smallest sigma-algebra on $X$ containing the topology $\mathcal{T}$.
$\endgroup$ $\begingroup$As answered by @bnd, Borel field is a special case of $\sigma$-field.
$\endgroup$Borel field is the smallest $\sigma$-field containing all open sets and due to complementation it contains all closed sets as well. And for openness and closedness we need a topology. Thus Borel $\sigma$-field is a field on a topological space which is smallest $\sigma$-field containing all open sets and closed sets.
