# existence of the Lebesgue measure

###### Theorem (Lebesgue).

Let $\mathcal{B}$ 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 $(\mathbb{R},\mathcal{B})$ satisfying

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

for all real numbers $a.

Title existence of the Lebesgue measure ExistenceOfTheLebesgueMeasure 2013-03-22 18:33:12 2013-03-22 18:33:12 gel (22282) gel (22282) 4 gel (22282) Theorem msc 28A12 msc 26A42 LebesgueMeasure Measure CaratheodorysExtensionTheorem