Borel measurable

In mathematics, specifically 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 additional restrictions on the measure, as described below.

Let ${\displaystyle X}$ be a locally compact Hausdorff space, and let ${\displaystyle {\mathfrak {B}}(X)}$ be the smallest σ-algebra that contains the open sets of ${\displaystyle X}$; this is known as the σ-algebra of Borel sets. A Borel measure is any measure ${\displaystyle \mu }$ defined on the σ-algebra of Borel sets. Some authors require in addition that ${\displaystyle \mu }$ is locally compact, meaning that ${\displaystyle \mu (C)<\infty }$ for every compact set ${\displaystyle C}$. If a Borel measure ${\displaystyle \mu }$ is both inner regular and outer regular, it is called a regular Borel measure (some authors also require it to be tight). If ${\displaystyle \mu }$ is both inner regular and locally finite, it is called a Radon measure.

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