All those rectangles

This result admits extension to higher dimensions if we take balls or cubes as substitutes for intervals of the real line, and it provides a quantitative version of the so-called fundamental theorem of calculus within the theory of Lebesgue, which is usually quoted as Lebesgue’s differentiation theorem. The original proof of Hardy and Littlewood was based on the properties of rearrangements of measurable functions, and was expressed in the picturesque language of cricket. But it can also be obtained from elementary geometrical considerations.