# Difference between Borel and sigma fields

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:

1. 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:
2. 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?

• Borel $\sigma$-field is the smallest $\sigma$-field that contains all open sets. $\sigma$-field is just a collection of subsets that meet the conditions you've listed Commented Oct 6, 2020 at 2:55
• @MoneyBall Sorry. I still don't understand the difference between them. Do you mind to give some example? Commented Oct 6, 2020 at 3:00

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.
• @YokJyeTang I don't understand what you mean by Borel field. Borel $\sigma$-field is important in measure theory and Lebesgue integration theory since we want to deal with sets that are measurable, and Borel $\sigma$-field contains pretty much all open sets you can think of. The sets that are not in Borel $\sigma$-field are often pathological and you often don't see them in practice. Commented Oct 6, 2020 at 3:11
• @YokJyeTang $\mathcal{B}$ is the Borel $\sigma$-field Commented Oct 6, 2020 at 3:15