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There are several equivalent ways to define a measurable subset of $\mathbb{R}$.

One way is to start with the Lebesgue outer measure and then restrict it to the set of subsets satisfying the Caratheodory criterion, namely the set of all sets $E$ such that $m^*A = m^*(A \cap E) + m^*(A \cap E^c)$ for all $A \subset \mathbb{R}$. This is the approach in H. L. Royden's Real Analysis.

Another way is to first define a positive linear functional on $C_c(\mathbb{R})$ and then apply the Riesz representation theorem on locally compact Hausdorff spaces. This is the approach in Rudin.

Another way is to start with a premeasure on an algebra of sets and then extend it to a measure on a $\sigma$-algebra. Yet another way mentioned here (which also happens to be the inspiration for this question) is to first define a subset with a finite outer measure to be measurable if its outer measure equals its inner measure. Then, for a general subset we take intersections with measurable subsets and define things the way they're supposed to be. This approach is taken in Frank Jones' book.

With all these different methods, it makes me wonder what Lebesgue's original approach was to define a measurable set. I haven't come across any textbook on Measure Theory/Real Analysis that points out what was Lebesgue's method.

So, which construction of measure did Lebesgue use, and what definition did he use for a measurable set?

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Lebesgue used what Caratheodory called the outer measure. Fermat's library has an annotated translation of Lebesgue's 1901 paper On a Generalization of the Definite Integral, he is much briefer than the subsequent elaborations:

"Let us consider a set of points of $(a, b)$; one can enclose in an infinite number of ways these points in an enumerably infinite number of intervals; the infimum of the sum of the lengths of the intervals is the measure of the set. A set $E$ is said to be measurable if its measure together with that of the set of points not forming $E$ gives the measure of $(a, b)$."

That is it. The translator remarks that "Lebesgue is defining $E$ to be measurable if $m^*(E) + m^*((a, b)\cap E^c)=b-a$", which is the "Caratheodory criterion" restricted to intervals.

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See :

$m(E)$ is a nonnegative measure on bounded sets $E$, such that:

(1) $m(E) \ne 0$ for some set $E$;

(2) $m(E + a) = m(E)$ for every real number $a$;

(3) if $E_n$ are pairwise disjoint, then $m(\bigcup E_n)=\sum m(E_N)$.

See also : G.T.Q. Hoare and N.J.Lord, "Intégrale, longueur, aire" the Centenary of the Lebesgue Integral, The Mathematical Gazette, Vol. 86 (2002).

And see also : David Bressoud, A radical approach to Lebesgue's theory of integration, Cambridge (2008), Ch.5 The Development of Measure Theory, on Borel and Lebesgue.

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  • $\begingroup$ What size is the collection in item (3)? $\endgroup$ – Mikhail Katz Apr 22 '18 at 8:34

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