The Jordan outer measure $J^*(E)$ of a set $E\subseteq \mathbb{R}$ is defined as infimim of $\sum_{i=1}^n (b_i-a_i)$ where $(a_i,b_i)$ are open intervals whose union contains $E$. The Jordan inner measure $J_*(E)$ of a set $E\subseteq \mathbb{R}$ is defined as supremum of
$\sum_{i=1}^n (b_i-a_i)$ where $(a_i,b_i)$ are open intervals, whose union is contained in $E$. A set is $E$ Jordan measurable if $J^*(E)=J_*(E)$.
Lebesgue measure of a set $E\subseteq \mathbb{R}$ is defined in a similar way by defining Lebesgue outer measure and inner measure, where the sums/unions in above definition are allowed to be countable.
Question:
What properties of functions can be characterized by the Lebesgue measure but not the Jordan measure?
(I want a motivation of Lebesgue measure with some drawback/disadvantages of Jordan measure. I didn't find theory of Jordan measure in many books of Measure theory, although it was a motivational point towards development of Lebesgue measure and Integration.)
Best Answer
Some drawbacks (assumed for convenience to be sited in $\mathbb{R}^n$): Not all compact sets are Jordan measurable, e.g. certain generalised Cantor sets in $\mathbb{R}$ with positive Lebesgue measure; no characterisation of Riemann integrable functions; assigns non-measurability to too many sets ("$A$ is Jordan measurable iff $\partial A$ has Jordan measure $0$") thereby making it a less discriminating estimate of the "size" of sets than Lebesgue measure; lacks the powerful integral convergence theorems of Lebesgue theory.
Nevertheless, see the very interesting Multidimensional Analysis vol.2 by Duistermaat & Kolk, which pushes the Jordan theory quite far and includes the Arzela integral theorem as a practical alternative for the dominated convergence theorem.