Lebesgue Measure

In measure theory, the Lebesgue measure is a way to extend the classical notions of length, area, and volume (n = 1, 2, 3 respectively) to n-dimensional Euclidean space. It is the standard way of assigning measure to subsets of n-dimensional Euclidean space. 
== Definition ==
For any subset A \subseteq \mathbb{R}, the Lebesgue outer measure, \lambda^*(A), with the length of an interval a,b given by \ell(I) = b-a, is defined as
\lambda^*(A) = \inf \left\{ \sum_n \ell(I_n) : (I_n)_n ~\text{is a countable collection of intervals s.t.}~ A \subseteq \bigcup_n I_n \right\}
For any subset A \subseteq \mathbb{R}^n, fix an n and define a box in \mathbb{R}^n as B = \textstyle \prod_{i=1}^n [a_i, b_i] \displaystyle where b_i \geq a_i, with its volume defined as vol(B) = \textstyle \prod_{i=1}^n (b_i - a_i) \displaystyle.Note that the box is defined as a set, so that the first product is the Cartesian product, where its volume is a real number so that the product is the standard numerical product. Then, the Lebesgue outer measure,\lambda^*(A), is defined as 
\lambda^*(A) = \inf \left\{ \sum_n vol(B_n) : (B_n)_n ~\text{is a countable collection of boxes s.t.}~ A \subseteq \bigcup_n B_n \right\}
Note: the intervals (I_n)_n can be open, closed, half open, etc. doesn't matter. I think the same holds for the boxes?
== Expand this entry ==
Sources to expand this entry:
* https://e.math.cornell.edu/people/belk/measuretheory/LebesgueMeasure.pdf
* https://people.math.harvard.edu/~shlomo/212a/11.pdf