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If this set does not have zero Lebesgue measure, then by countable additivity of the measure there is at least one such n so that X1/n does not have a zero measure. Thus there is some positive number c such that every countable collection of open intervals covering X1/n has a total length of at least c. In particular this is also true for every such finite collection of intervals. This remains true also for X1/n less a finite number of points (as a finite number of points can always be covered by a finite collection of intervals with arbitrarily small total length).

For every partition of [a, b], consider the set of intervals whose interiors include points from X1/n. These interiors consist of a finite open cover of X1/n, possibly up to a finite number of points (which may fall on interval edges). Thus these intervals have a total length of at least c. Since in these points f has oscillation of at least 1/n, the infimum and supremum of f in each of these intervals differ by at least 1/n. Thus the upper and lower sums of f differ by at least c/n. Since this is true for every partition, f is not Riemann integrable.

We now prove the converse direction using the sets Xε defined above.[9] ・・
(引用終り)

＜補足＞
１）Proofで、Darboux integral https://en.wikipedia.org/wiki/Darboux_integral
　に持ち込むのが、定石のようです
　Darboux integralを見ると、"supとinf" https://manabitimes.jp/math/1140 高校数学の美しい物語 sup(上限)とinf(下限)の意味，max・minとの違い 2022/08/15
　この"supとinf"は、数学では常用の手筋ですね
２）上記”One direction”は、「不連続の点の集合がmeasure zero→Darboux integral 不可」ですね
　背理法ですね。”If this set does not have zero Lebesgue measure, then by countable additivity of the measure there is at least one such n so that X1/n does not have a zero measure.”
３）そして、測度0でなければ
　”Thus these intervals have a total length of at least c. Since in these points f has oscillation of at least 1/n, the infimum and supremum of f in each of these intervals differ by at least 1/n. Thus the upper and lower sums of f differ by at least c/n. Since this is true for every partition, f is not Riemann integrable.”
　を導きます
　このとき、”oscillation ”https://en.wikipedia.org/wiki/Oscillation_(mathematics)
　を使って、不連続の評価をしています。（不勉強で、”oscillation ”は初見でしたが、面白いですね。定石かも）
　”For every positive ε, Let Xε be the set of points in [a, b] with oscillation of at least ε.”に続きます

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