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 .