Emile Borel

I wish to clarify two points to which my attention has been called by a correspondent. Both occur on page 500. (1) The passage on the covering theorem in the second paragraph on page 500 is open to misunderstanding. The omission from the statement of the theorem in his thesis of the condition that the covering is enumerable, which is implicit in the proof by transfinite induction and suffices for the applications made, was clearly a mere oversight on Borel's part which was made good in the Leçons . No question of priority is involved. (2) The second point arises in the last paragraph on page 500 and concerns Borel's proof of the consistency of his definition of measure by means of the covering theorem (for an enumerable covering). This is not well known. It was not published until 1912 when it appeared in the memoir [ 134 ] on integration, reprinted, under a different title, as Note VI in the second and subsequent editions of the Leçons . In the first edition Borel had contented himself with the assertion that consistency is assured by the covering theorem and there seems no reason to doubt that the proof he had then was that published in 1912. It depends on using a mixed covering of “subtractive” as well as “additive” intervals as they arise in the construction of a Borel set. The first proof of consistency to be published, however, was due to Lebesgue (1902) and depends upon his notion of outer measure which is foreign to Borel's theory. Lebesgue's proof is much the simpler, but Borel's showed that his theory is self‐contained.