On Riemann sums and Lebesgue integrals

Call a sequence of positive integers(m _k ) _k=1 ^∞ a chain ifm _k devidesm _k+1 and that it has dimensiond if it is a subset of the set of least common multiples ofd chains. In this paper we give a new and elementary proof that iff∈L(logL)^d?1([0, 1)) and(m _k ) _k=1 ^∞ is of dimensiond then $$mathop {lim }limits_{N to infty } frac{1}{N}sumlimits_{n = 1}^N {fleft( {left{ {x + frac{n}{{m_N }}} right}} right)} = int_X {fdmu , a.e.,} $$ with respect to Lebesgue measure. This result was first proved byL. Dubins andJ. Pitman [2] using martingale theory.