3 · Measure what happens when boxes merge

A fine-stage measure must stay within its size limit after its boxes are merged at a coarser stage. This chart calculates the change for equal weights.

Try this. Choose an exponent above one and merge several boxes. Then move to one or below. The ratio is now at most one, so the same size limit works at both stages.

For equal weights, the ratio is n raised to one minus one over p. Equal weights give the worst growth above one, while subadditivity controls all weights at or below one.