Can anyone give me explains to me what is the difference between the Borel field and the sigma field? Definition of Sigma field with the uncountable sample space:
- All subset of the sample space that can be obtained by countable many intersections and unions of the interval of the form [x1,x2] with x1 <= x2 Definition of Borel field with the uncountable sample space:
- All subset of the sample space that can be obtained by countable many intersections and unions of open intervals of the form (a,b) with a <= b.
From here, isn't it the difference between them is just from closed and open forms? Then, why we need Borel field? And, why it's important?
Given the example of sample space: [0,1], can someone construct an example of sigma field and borel field?
σ-field is a collection of sets that is closed under countable unions, countable intersections, and complements. Borel σ-field is the smallest σ-field that contains all open sets.
Given a space $\Omega = (0,1)$, $\mathcal{A} = \{ \Omega, \emptyset\}$ is trivially a $\sigma$-field (the intersection is the empty set, union is $\Omega$, and both are complements of each other), but $\mathcal{A}$ is not a Borel $\sigma$-field since it doesn't contain any open sets in $\mathbf{R}$.
Let $\mathcal{B}$ be the Borel $\sigma$-field. The term "smallest" means that for any $\mathcal{S}$ that contains all open sets, then $\mathcal{B} \subset \mathcal{S}$. In fact, we know there exists a smallest set since we can define $\mathcal{B} = \bigcap \mathcal{S_n}$, where $\mathcal{S}_n$ is a $\sigma$-field that contains all open sets.