Borel set

In mathematics, the Borel sets of a topological space are a particular class of "well-behaved" subsets of that space. For example, whereas an arbitrary subset of the real numbers might fail to be Lebesgue measurable, every Borel set of reals is universally measurable. Which sets are Borel can be specified in a number of equivalent ways. Borel sets are named after Émile Borel.

The most usual definition goes through the notion of a σ-algebra, which is a collection of subsets of a topological space ${\displaystyle X}$ that contains both the empty set and the entire set ${\displaystyle X}$, and is closed under countable union and countable intersection.

Then we can define the Borel σ-algebra over ${\displaystyle X}$ to be the smallest σ-algebra containing all open sets of ${\displaystyle X}$. A Borel subset of ${\displaystyle X}$ is then simply an element of this σ-algebra.

Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.

In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces.

In the case that ${\displaystyle X}$ is a metric space, the Borel algebra in the first sense may be described generatively as follows.

For a collection ${\displaystyle T}$ of subsets of ${\displaystyle X}$ (that is, for any subset of the power set ${\displaystyle {\mathcal {P}}}$${\displaystyle (X)}$ of ${\displaystyle X}$), let

Now define by transfinite induction a sequence ${\displaystyle G^{m}}$, where ${\displaystyle m}$ is an ordinal number, in the following manner: