For example, to study an arbitrary subset of A of Euclidean space, one may instead replace it by a union of dyadic cubes of a particular size that cover the set.
Most notable appearances of dyadic cubes include the Whitney extension theorem and the Calderón–Zygmund lemma.
In practice, however, letting Δk be the set of cubes of sidelength 2k or 2−k is a matter of preference or convenience.
The Hardy–Littlewood maximal inequality states that for an integrable function f, for λ > 0 where Cn is some constant depending only on dimension.
The last condition says that the area near the boundary of a "cube" Q in Δ is small, which is a property taken for granted in the Euclidean case although is very important for extending results from harmonic analysis to the metric space setting.