Definition:Lebesgue Measure

Lebesgue Measure on the Reals
For a given set $$S \in \mathbb{R}$$, let $$\left\{{I_n}\right\}$$ be a countable set of open intervals such that

$$S \subseteq \bigcup I_n$$

For the set of all subsets $$\mathcal{P} \left({\mathbb{R}}\right)$$ of the reals $$\mathbb{R}$$, construct a function $$m^*:\mathcal{P} \left({\mathbb{R}}\right) \to \mathbb{R}_+$$ as

$$m^*(S) = \inf_{\left\{{I_n}\right\} :S \subseteq \cup I_n} \sum l (I_n) \ $$

where the infimum ranges over all such sets $$\left\{{I_n}\right\} \ $$, and $$l(I_n) \ $$ is the length of the interval.

For the set $$\mathfrak {M} \ $$ of measurable sets, the Lebesgue Measure $$m:\mathfrak {M} \to \mathbb{R}_+$$ is defined as

$$m(S) = m^*(S) \ $$