Kakutani's Equidistribution Theorem

Kakutani's equidistribution theorem states that the largest-first refinement defining Kakutani partitions produces a sequence of asymptotically uniformly distributed finite point sets in the unit interval (Kakutani 1976, Smilansky 2020).

Fix and begin with . At each stage, split every longest current interval into two intervals in the left-to-right ratio , leaving shorter intervals unchanged. Let contain one arbitrarily chosen point in the interior of each interval after refinement stages. Then

for every , where is the cardinal number of . Equivalently, for every continuous function ,

The index counts refinement stages, and the finite point sets are sampled separately.

For , the intervals at stage all have length . Choosing their midpoints gives , so the average of over is the midpoint Riemann sum for its integral on .