2 · Integrate at a coarse or fine stage

A continuous integer-valued function is constant on the boxes of some finite stage. We can repeat its values on finer boxes without changing the function.

Try this. Move the reading stage. The bars split into finer bars, but the weighted total stays fixed because the child weights add up to their parent weight.

This stage independence makes integration against a compatible measure well defined. The tests compute the same integral at coarse and fine stages and require exact equality.