Jordan Measure and Lebesgue Measure

The Jordan outer measure $$J^*(E)$$ of a set $$E\subseteq \mathbb{R}$$ is defined as infimum 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.)

• A classic motivational example is the set of rational numbers within $[0,1]$ - it is not Jordan measurable, but it has Lebesgue measure zero. Commented Aug 7, 2014 at 8:33

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.
• Regarding "no characterisation of Riemann integrable functions", Jordan measure can be used to give a characterization and in fact the pre-Lebesgue characterizations used this: A bounded function $f:[a,b] \rightarrow {\mathbb R}$ is Riemann integrable if and only if for each $\epsilon > 0$ the set of points at which the oscillation of $f$ is greater than or equal to $\epsilon$ has Jordan measure zero. Indeed, it is this characterization that H. J. S. Smith used in showing that certain functions were not Riemann integrable in his 1875 paper where the first Cantor set (essentially) appeared. Commented Aug 7, 2014 at 15:09