Consistent first-return Riemann sums for Lebesgue integrals

U. B. Darji and M. J. Evans 1] showed previously that it is possible to obtain the integral of a Lebesgue integrable function on the interval 0,1] via a Riemann type process, where one chooses the selected point in each partition interval using a first-return algorithm based on a sequence {x _n} which is dense in 0,1]. Here we show that if the same is true for every rearrangement of {x _n}, then the function must be equal almost everywhere to a Riemann integrable function. This revised version was published online in June 2006 with corrections to the Cover Date.