existence of the Lebesgue measure

Theorem (Lebesgue).

Let $\mathrm{B}$ be the Borel $\sigma$-algebra (http://planetmath.org/BorelSigmaAlgebra) on the real number line. Then, there is a unique measure $\mu$ on the measurable space $\mathrm{(}\mathrm{R}\mathrm{,}\mathrm{B}\mathrm{)}$ satisfying

$$\mu \left((a,b)\right)=b-a$$

for all real numbers .

Title	existence of the Lebesgue measure
Canonical name	ExistenceOfTheLebesgueMeasure
Date of creation	2013-03-22 18:33:12
Last modified on	2013-03-22 18:33:12
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	4
Author	gel (22282)
Entry type	Theorem
Classification	msc 28A12
Classification	msc 26A42
Related topic	LebesgueMeasure
Related topic	Measure
Related topic	CaratheodorysExtensionTheorem