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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.)

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

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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.

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    $\begingroup$ 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. $\endgroup$ Commented Aug 7, 2014 at 15:09
  • $\begingroup$ @DaveL.Renfro Much obliged for the correction. $\endgroup$
    – InTransit
    Commented Aug 8, 2014 at 6:22
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Question: What properties of functions can be characterized by the Lebesgue measure but not the Jordan measure?

Answer: Jordan measure(which actually is NOT measure, though the name says “measure”. So confusing term) is very weak concept coampred to Lebesgue measure. Firstly, Jordan measure of a set E in R^d always is to be +∞ if E is unbounded. This is very unsatisfactory because it implicitly says integral with Jordan measure always has to deal with unbounded set as +∞. So whatever underlying sets and functions you try to integrate, the value can highly be +∞ as long as underlying set or range of function contains unbounded set. You can see this is why we were very nervous about doing "irregular integrals". Secondly, even if E is bounded closed set or compact, there’s an counterexample of non-Jordan measurable set. The famous one is (Q∩[0,1]). Judging from (at least) the two drawbacks of Jordan measure, one can intuitively say “Jordan measure” is very fragile for underlying sets with 1. some kinds of unboundeness 2. very intricated like fractal to define an integral theory which is stronger enough to integrate functions equipped with the above properties.

The theory of Lebesgue integral is sometimes called the completion of “Remmal integral”. I highly recommend you to read “An introduction to Measure theory” written by Terrence Tao.

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