Jordan measure

Extension to more complicated sets

Formally, for a bounded set ${\displaystyle B,}$ define its inner Jordan measure as $${\displaystyle m_{*}(B)=\sup _{S\subseteq B}m(S)}$$ and its outer Jordan measure as $${\displaystyle m^{*}(B)=\inf _{S\supseteq B}m(S)}$$ where the infimum and supremum are taken over simple sets ${\displaystyle S.}$ The set ${\displaystyle B}$ is said to be a Jordan measurable set if the inner measure of ${\displaystyle B}$ equals the outer measure. The common value of the two measures is then simply called the Jordan measure of ${\displaystyle B}$. The Jordan measure is the set function that sends Jordan measurable sets to their Jordan measure.