Borel field

Definition

Suppose $A_1,A_2,.. \in \mathcal{F}$, where $\mathcal{F}$ is a field, then it is a Borel field if all unions and intersections are also in $\mathcal{F}$.

Alternatively: the Borel σ-algebra is the σ-algebra generated by the open sets.

Notes

• the open sets of a topological space $(X,\mathcal{F})$ general a σ-algebra which is the Borel σ-algebra of the space

See also

• topology
• topological space
• measurable space

References

1. https://mathworld.wolfram.com/BorelField.html
2. https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html
3. https://mathworld.wolfram.com/BorelSigma-Algebra.html
4. https://en.wikipedia.org/wiki/Borel_set
5. https://math.stackexchange.com/questions/2323671/is-borel-field-different-from-sigma-field
6. https://math.stackexchange.com/questions/1080473/borel-sigma-algebra-definition
7. https://math.stackexchange.com/questions/1330649/difference-between-topology-and-sigma-algebra-axioms
8. https://mathoverflow.net/questions/31603/why-do-probabilists-take-random-variables-to-be-borel-and-not-lebesgue-measura/31724#31724
9. https://mathoverflow.net/questions/87838/is-every-sigma-algebra-the-borel-algebra-of-a-topology