Lebesgue measure

Definition

Consider $n$ -dimensional closed rectangle with sides oriented parallel to coordinate axes ('rectangle'), which is a subset $R \subset \mathbb{R}^n$ of form

R = [a_1,b_1] \times [a_2, b_2] \times ... \times [a_n,b_n]

where $-\infty < a_i \leq b_i < \infty$ for $i = 1,...,n$ .

Volume $\mu(R)$ of $R$ is

\mu(R) = (b_1-a_1)(b_2-a_2)\dots(b_n-a_n)

(corresponding to length or area respectively for $n=1, 2$ )

Denote collection of all $n$ -dimensional rectangles by $\mathcal{R}(\mathbb{R}^n)$ (or simply $\mathcal{R}$ ), then $R \mapsto \mu(R)$ defines a map $\mu : \mathcal{R}(\mathbb{R}^n) \to [0,\infty)$

Define the outer Lebesgue measure $\mu^*(E)$ of subset $E \subset \mathbb{R}^n$ , or outer measure for short, as

\mu^*(E) = \inf \left\{\sum_{i=1}^\infty \mu(R_i) : E \subset \cup_{i=1}^\infty R_i, R_i \in \mathcal{R}(\mathbb{R}^n)\right\}

#incomplete

Notes

for $n=1,2,3$ is simply length, area, volume respectively

References

https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch2.pdf
https://en.wikipedia.org/wiki/Lebesgue_measure
https://e.math.cornell.edu/people/belk/measuretheory/LebesgueMeasure.pdf