In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations...

algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular...

in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require...

are Borel isomorphic: there is a bijection from X to Y such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This...

of a Borel set in a Polish space. A is a Suslin set, the image of the Suslin operation. There is a Polish space Y {\displaystyle Y} and a Borel set B ⊆...

theorem Borel right process Borel set Borel summation Borel distribution Borel's conjecture about strong measure zero sets (not to be confused with Borel conjecture...

Moreover, every Borel set is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets. Any countable set of real numbers...

measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent...

of a non-complete Borel measure. The Borel measure is not complete. One simple construction is to start with the standard Cantor set K , {\displaystyle...

n-dimensional Euclidean space Rn is called a Borel regular measure if the following two conditions hold: Every Borel set B ⊆ Rn is μ-measurable in the sense of...