Dyadic cubes of all generations in R^n

Fix an integer $n\ge 1$. Let $\mathbb{Z}$ be the integers of The integers as equivalence classes of pairs of naturals with their order and ring operations, and read integer values inside $\mathbb{R}$ along the canonical embedding. For $k\in \mathbb{Z}$ and a function $m:n\to \mathbb{Z}$, the dyadic cube of generation $k$ and index $m$ is the half-open box (Half-open boxes in ${\mathbb{R}}^n$ and their volume) $Q_{k,m}:=\{x\in {\mathbb{R}}^n:m_i{2}^{-k}<x_i\le (m_i+1)2^{-k}\text{ for every }i<n\},$ where $2^{-k}$ is the integer power of Integer powers $a^m$ and the products $m_i{2}^{-k}$ are read in $\mathbb{R}$. The generation of the cube is $k$ and its side length is $2^{-k}$. Since $2^{-k}>0$ by Laws of integer exponents, the two endpoints of each coordinate interval satisfy $m_i{2}^{-k}<(m_i+1)2^{-k}$, so $Q_{k,m}$ is a nonempty half-open box with both parameters finite; assuming Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), its measure, computed from the half-open box measure theorem (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included), is the product of the $n$ equal side lengths $\mathrm{\mid }{Q}_{k,m}\mathrm{\mid }=(2^{-k}{)}^n=2^{-kn}\mathrm{.}$

Generations are indexed by all of $\mathbb{Z}$, not merely by the natural numbers: for $k>0$ the side length $2^{-k}$ is smaller than $1$, for $k=0$ it is $1$, and for $k<0$ the side length is $2^{\mathrm{\mid }k\mathrm{\mid }}>1$, so cubes larger than the unit cube occur. The cubes with $k\ge 0$ are exactly the generation-$k$ dyadic cubes of the measure-theoretic convention Dyadic cubes of generation $k$ in ${\mathbb{R}}^n$, whose generation index is a natural number; that convention is bounded above in size by the unit cube $k=0$. Such a grid does not suffice for a stopping-time decomposition at a small height: a maximal bad cube can then be the unit cube while the height is far below its average. The all-generations family above is the grid used by the dyadic maximal function and by the Calderón–Zygmund decomposition on this page; the next item proves that it partitions ${\mathbb{R}}^n$ at each generation, that each cube has exactly one ancestor of every coarser generation, and that two cubes are nested or disjoint. All parameters and the generation are determined by the cube as a set, by the parameter uniqueness recorded in Half-open boxes in ${\mathbb{R}}^n$ and their volume. The set construction is choice-free; only the stated identification with Lebesgue measure uses Countable Choice.