**Borel Measure**

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let *X* be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of *X*; this is known as the σ-algebra of Borel sets. Any measure *μ* defined on the σ-algebra of Borel sets is called a **Borel measure**. Some authors require in addition that *μ*(*C*) < ∞ for every compact set *C*. If a Borel measure *μ* is both inner regular and outer regular, it is called a **regular Borel measure**. If *μ* is both inner regular and locally finite, it is called a Radon measure. Note that a locally finite Borel measure automatically satisfies *μ*(*C*) < ∞ for every compact set *C*.

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### Famous quotes containing the word measure:

“Outward simplicity befits ordinary men, like a garment made to *measure* for them; but it serves as an adornment to those who have filled their lives with great deeds: they might be compared to some beauty carelessly dressed and thereby all the more attractive.”

—Jean De La Bruyère (1645–1696)