outer measure

Definition

For a set $A\subseteq \mathbb{R}$ $|A|=\inf \{{\sum }_{k=1}^{\infty }\ell (I_k)|{\bigcup }_{k=1}^{\infty }{I}_k\text{is an open covering of }A\}$ Where $\ell (I_k)$ is the length of the interval

Notes

The outer measure is NOT a measure on the 20231205121417-sigma_algebra the powerset of $\mathbb{R}$ because of the Vitali set.