Measures are used to formalize notions of length, area, and volume and expand the notions into more cases. There are many ways to define a measure functions, but they each satisfy the following properties.
We will define \ell(I_k) to be the length of the interval. For example \ell([a,b]) = b - a
A measure is a function from the measurable sets \mathscr{M} to the non-negative real numbers m \colon \mathscr{M} \to \mathbb{R}_{\geq 0} such that the following hold.
The measure of an interval is its length. Meaning the measure of every nonempty interval I is
m(I) = \ell(I)
Measure is translation invariant. Meaning that if E is a measurable set, than E + y = \{x+y | x \in E\} is also measurable and
m(E) = m(E + y)
- Measure is countably additive over a disjoint union of sets. Meaning that if \{E_k\}_{k=1}^\infty if a collection of disjoint measurable sets, then
m\left(\bigcup_{k=1}^\infty E_k \right) = \sum_{k=1}^{\infty}m(E_k)
In order to construct the Lebesgue measure can, we define a function m^* \colon \mathcal{P}(\mathbb{R}) \to \mathbb{R} called the outer measure. We then define a collection \mathcal{M} such that m^* restricted to \mathscr{M} fufills our requirements of being measure as defined above.
For any subset E \subseteq \mathbb{R} A Lebesgue outer measure on E is defined
m^*(E) = inf\left\{ \sum_{k=1}^{\infty} \ell(I_k) \right\}
Where E \subseteq \bigcup_{k=1}^\infty I_k
In English, the Lebesgue Outer Measure is composed of the least collection of intervals which cover E without overlapping. The total of the outer measure may overestimate E however, as the intervals may contain points that aren’t in E.
A set E \subseteq \mathbb{R} is measurable if and only if for any A \subseteq \mathbb{R}
m^*(A) = m^*(A \cap E) + m^*(A \cap E^c)