Theorem 1 (Lebesgue) Let $\mathcal{B}$ be the Borel $\sigma$ algebra on the real numberline. Then, there is a unique measure$\mu$ on the measurable space$(\mathbb{R},\mathcal{B})$ satisfying \begin{equation*} \mu\left( (a,b) \right) = b-a \end{equation*}for all real numbers $a<b$
"existence of the Lebesgue measure" is owned by gel.
