The outer measure and its applications: the Lebesgue measure

Abstract

An outer measure on a set R is a non-negative-valued (possibly infinite), monotone, countably sub-additive set-function, defined on all subsets of R, which vanishes on the empty set. If we denote it by m*, we have (a) \(m*\left( {\not 0} \right) = 0;\) (b) A ⊂ R, B ⊂ R, A ⊂ B ⇒ m*(A) ≤ m*(B); (c) \({A_n} \subset R{\kern 1pt} for{\kern 1pt} n = 1,2, \ldots , \Rightarrow m*\left( { \cup _{i = 1}^\infty {A_i}} \right) \leqslant \sum\nolimits_{i = 1}^\infty {m*\left( {{A_i}} \right).}\)