Borel Set

• categories: Measure Theory, Definition

A Borel set is any set that can be constructed from open sets in a topological space (typically $\mathbb{R}$ with its standard topology) through countable operations, such as unions, intersections, and complements.

Key Properties:

1. Closed under countable operations:
   • If $A_1,A_2,\dots$ are Borel sets, then ${\bigcup }_{n=1}^{\infty }{A}_n$, ${\bigcap }_{n=1}^{\infty }{A}_n$, and the complement $A^c$ are also Borel sets.
2. Generation from open sets:
   • The Borel $\sigma$-algebra is generated by open sets, meaning that any Borel set can be expressed in terms of open (or equivalently, closed) sets via countable unions, intersections, and complements.
3. Includes many common sets:
   • All open and closed sets are Borel sets.
   • Countable unions and intersections of open (or closed) sets are Borel sets.
   • Intervals such as $[a,b]$, $(a,b]$, and $(a,b)$ are Borel sets.

Intuition:

Borel sets form a broad class of sets that include many “well-behaved” subsets of $\mathbb{R}$ (such as intervals), but they also include much more complex sets that are still describable in terms of open sets through countable processes.

Importance:

Borel sets are fundamental in measure theory because they are the sets for which the Lebesgue measure is defined. The Borel $\sigma$-algebra allows us to extend the notion of measure beyond simple geometric objects like intervals to more complicated sets.