Elementary volume is a sigma-finite premeasure on the algebra of elementary sets

Given: A natural number $n\ge 1$, the algebra ${\mathcal{E}}_n$ with elementary volume ${\mu }_0$, and a pairwise disjoint sequence $(A_k{)}_{k\in \mathbb{N}}$ in ${\mathcal{E}}_n$ whose union $A$ lies in ${\mathcal{E}}_n$.

${\mathcal{E}}_n$ is an algebra of subsets of ${\mathbb{R}}^n$, it contains every half-open box, and it is closed under intersection of two members and under difference (The elementary sets form an algebra of subsets of ${\mathbb{R}}^n$ containing every half-open box).

For every $n\ge 1$, there is exactly one function ${\mu }_0:{\mathcal{E}}_n\to [0,+\mathrm{\infty }]$ whose value at $A$ is the sum of the volumes of the members of any presentation of $A$ by a finite list of pairwise disjoint half-open boxes; it satisfies ${\mu }_0(\mathrm{\varnothing })=0$ and ${\mu }_0(B)=\mathrm{vol}(B)$ for every half-open box $B$ (The sum of the volumes of a disjoint box decomposition of an elementary set does not depend on the decomposition).

Elementary volume is finitely additive on pairwise disjoint elementary sets, monotone, and finitely subadditive (Elementary volume is finitely additive, monotone and finitely subadditive on the elementary algebra).

$A^{+\delta }$ is an elementary set, it is determined by $A$ and $\delta$ alone, it contains $A$, and every point of $A$ is an interior point of $A^{+\delta }$ in $({\mathbb{R}}^n,d_2)$ (Every elementary set is squeezed in volume between a compact subset and an elementary set whose interior contains it, claim 1).

For every real $\epsilon >0$ there is $m\in \mathbb{N}$ with ${\mu }_0\left(A^{+1\mathrm{/}(m+1)}\right)\le {\mu }_0(A)+\epsilon$ (Every elementary set is squeezed in volume between a compact subset and an elementary set whose interior contains it, claim 2).

If ${\mu }_0(A)<+\mathrm{\infty }$, then for every real $\epsilon >0$ there are an elementary set $A^{\mathrm{\prime }}$ and a compact set $K\subseteq {\mathbb{R}}^n$ with $A^{\mathrm{\prime }}\subseteq K\subseteq A$ and ${\mu }_0(A)\le {\mu }_0(A^{\mathrm{\prime }})+\epsilon$ (Every elementary set is squeezed in volume between a compact subset and an elementary set whose interior contains it, claim 3).

For a nonempty box $\mathrm{vol}(B):=+\mathrm{\infty }$ when $a_i=-\mathrm{\infty }$ or $b_i=+\mathrm{\infty }$ for some $i<n$, and $\mathrm{vol}(B):={\prod }_{i<n}(b_i-a_i)$ when every $a_i$ and every $b_i$ is real; $B(a,b):=\{x\in {\mathbb{R}}^n:a_i<x_i\le b_i\text{ for every }i<n\}$; and $(u,v{]}^n:=B(\mathbf{u},\mathbf{v})$ (Half-open boxes in ${\mathbb{R}}^n$ and their volume).

A subset $E\subseteq {\mathbb{R}}^n$ is an elementary set when there are a natural number $m$ and a list $B_0,\dots ,B_{m-1}$ of half-open boxes with $E={\bigcup }_{j<m}{B}_j$ (Elementary sets: the finite unions of half-open boxes in ${\mathbb{R}}^n$).

A premeasure on an algebra ${\mathcal{A}}_0$ vanishes at the empty set and is countably additive whenever a disjoint sequence in ${\mathcal{A}}_0$ has its union in ${\mathcal{A}}_0$; it is sigma-finite if there is a sequence $(P_n)$ in ${\mathcal{A}}_0$ with $X={\bigcup }_n{P}_n$ and ${\mu }_0(P_n)<+\mathrm{\infty }$ for every $n$ (Premeasures on algebras of sets).

The nonnegative extended sum of a sequence in $[0,+\mathrm{\infty }]$ is ${\sum }_{k=0}^{\mathrm{\infty }}{a}_k:={\sup }_{n\in \mathbb{N}}{s}_n$, the supremum of its nondecreasing partial sums (Series in the nonnegative extended real line).

$A$ is a compact subset of $X$ if and only if for every set $I$ and every family $(U_i{)}_{i\in I}$ of open subsets of $X$ with $A\subseteq {\bigcup }_{i\in I}{U}_i$ there are $n\in \mathbb{N}$ and indices $i_0,\dots ,i_n\in I$ with $A\subseteq U_{i_0}\cup \cdots \cup U_{i_n}$, or else $A=\mathrm{\varnothing }$ (A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, claim 3; Open cover, subcover, compact metric space, and compact subset of a metric space).

The interior $\mathrm{int}(A)$ is open, and it is the largest open subset of $A$ (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

Every nonempty subset $S\subseteq \mathbb{N}$ has a least element (The well-ordering principle).

If $\mathrm{\mid }r\mathrm{\mid }<1$ then ${\sum }_{k=0}^{\mathrm{\infty }}{r}^k=1\mathrm{/}(1-r)$; in particular ${\sum }_{k=0}^{\mathrm{\infty }}{2}^{-k}=2$ (For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges).

Every complete ordered field $F$ is Archimedean: for every $x\in F$ there is a natural number $n\ge 1$ with $x<n\cdot 1_F$ (Every complete ordered field is Archimedean).

For sequences of reals, ${\sum }_{k<n}(a_k+b_k)={\sum }_{k<n}{a}_k+{\sum }_{k<n}{b}_k$; if $a_k\le b_k$ for all $k<n$ then ${\sum }_{k<n}{a}_k\le {\sum }_{k<n}{b}_k$; and if $a_k\ge 0$ for all $k<n$ then ${\prod }_{k<n}{a}_k\ge 0$, with ${\prod }_{k<n}{a}_k>0$ when every $a_k>0$ (Laws of finite sums and finite products, claims 1, 4 and 6).

Finite sums and finite products of a sequence of reals are defined by the recursions ${\mathrm{\Sigma }}_0=0$, ${\mathrm{\Sigma }}_{\sigma (n)}={\mathrm{\Sigma }}_n+a_n$ and ${\mathrm{\Pi }}_0=1$, ${\mathrm{\Pi }}_{\sigma (n)}={\mathrm{\Pi }}_n\cdot a_n$ (Finite sums and finite products, by recursion).

For $a,b\in \bar{\mathbb{R}}$, $a+b:=+\mathrm{\infty }$ when $a=+\mathrm{\infty }$ and $b\mathrm{\ne }-\mathrm{\infty }$, or $b=+\mathrm{\infty }$ and $a\mathrm{\ne }-\mathrm{\infty }$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined).