Geodesic
A Characterization of the Lebesgue Integral
Canadian mathematical bulletin, Tome 20 (1977) no. 3, pp. 353-357

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The first general integrability criterion is due to Riemann. He observed that a necessary and sufficient condition for a function f(x) on [a, b] to be integrable in his sense is that for each ε > 0 there exists a δ > 0 so that where {x 0, x 1, ..., xn } is a partition of [a, b] with diameter less than δ (and ω(f, I) denotes the oscillation of f on the interval I). 
Thomson, B. S. A Characterization of the Lebesgue Integral. Canadian mathematical bulletin, Tome 20 (1977) no. 3, pp. 353-357. doi: 10.4153/CMB-1977-053-3
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     title = {A {Characterization} of the {Lebesgue} {Integral}},
     journal = {Canadian mathematical bulletin},
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     year = {1977},
     volume = {20},
     number = {3},
     doi = {10.4153/CMB-1977-053-3},
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