Euclidean Space and Lebesgue Measure: Standing Notation

This setting fixes the standing notation used by results of real analysis and measure theory on Euclidean space. It introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it.

1. (Numbers, sequences, and finite index sets)¶ $\mathbb{N}$ denotes the natural numbers, $\mathbb{Z}$ the integers, and $\mathbb{R}$ the real numbers, carrying the addition, multiplication and order $\le$ of their ordered field structure together with the notation fixed there: $a<b$ for the associated strict order, $a-b$ for $a+(-b)$, and $a^{-1}$ for the multiplicative inverse of a nonzero $a$. We write $|\cdot|$ for the absolute value on $\mathbb{R}$, and call $a\in\mathbb{R}$ positive if $0<a$ and nonnegative if $0\le a$. Powers with natural exponent are those of Natural Number Power of an Element of a Field, and for nonzero $a\in\mathbb{R}$ and $k\in\mathbb{N}$ we write $a^{-k}$ for the multiplicative inverse of $a^{k}$. A sequence in a set $X$ is written $(x_{m})_{m\in\mathbb{N}}$, and convergence of a sequence of real numbers is as defined there. Finiteness and countability of sets are as defined there, $|X|$ denotes the number of elements of a finite set $X$, $[m]$ denotes the initial segment of $\mathbb{N}$ determined by $m\in\mathbb{N}$, and $X^{q}$ denotes the set of $q$-tuples in $X$ for $q\in\mathbb{N}$. Least upper bounds and greatest lower bounds of subsets of $\mathbb{R}$ are those of Upper Bound and Least Upper Bound and Lower Bound and Greatest Lower Bound.

2. (Euclidean space, its metric and its topology)¶ Throughout, $q$ denotes a natural number with $1\le q$; the notation of this clause and of clauses 3 to 5 is introduced for every such $q$ simultaneously, and a result adopting this setting uses it for whichever dimensions it names. Thus $\mathbb{R}^{q}$ is Euclidean space, a real vector space under the sum $x+y$ of points and the scalar multiple $\mu x$ by $\mu\in\mathbb{R}$; $x\cdot y$ and $x-y$ denote the dot product and difference of $x,y\in\mathbb{R}^{q}$, and $\lVert\,\cdot\,\rVert$ the Euclidean norm. The Euclidean distance $d_{E}$ is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$ and satisfies $d_{E}(x,y)=\lVert x-y\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so $(\mathbb{R}^{q},d_{E})$ is a metric space. The subsets open in $(\mathbb{R}^{q},d_{E})$ form a topology by Metric Open Sets Form a Topology and coincide with the subsets open in the Euclidean sense by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$; such a set is called open without further qualification. Closedness, interior and closure are taken in that topology, boundedness refers to $d_{E}$, and compactness to that topology. For $x\in\mathbb{R}^{q}$ and positive $r\in\mathbb{R}$, $B(x,r)$ and $\bar{B}(x,r)$ denote the open ball and the closed ball of $(\mathbb{R}^{q},d_{E})$ with centre $x$ and radius $r$.

3. (The constant $\sigma_{q}$)¶ We write $\sigma_{q}=\lVert(1,\dots,1)\rVert$ for the norm of the point of $\mathbb{R}^{q}$ all of whose coordinates equal $1$. By claims 1 and 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ the number $\sigma_{q}$ is positive and satisfies $\sigma_{q}^{2}=q$; it is the constant introduced under the same name in Uniform Grids on a Half-Open Box and Grid Hulls of a Compact Set in $\mathbb{R}^n$.

4. (Borel sets)¶ $\mathcal{B}(\mathbb{R}^{q})$ denotes the Borel $\sigma$-algebra of $\mathbb{R}^{q}$, a $\sigma$-algebra on $\mathbb{R}^{q}$. By The Borel $\sigma$-Algebras of Euclidean Space and of the Euclidean Metric Coincide it is the $\sigma$-algebra generated by the open subsets of $(\mathbb{R}^{q},d_{E})$; in particular every open subset of $\mathbb{R}^{q}$ belongs to $\mathcal{B}(\mathbb{R}^{q})$, and so does every closed subset, since a closed set is the complement of an open one and a $\sigma$-algebra contains the complement of each of its members by Sigma-Algebra and Measurable Space.

5. (Lebesgue measure and null sets)¶ $\lambda_{q}$ denotes Lebesgue measure on $\mathcal{B}(\mathbb{R}^{q})$, a measure with values in $[0,\infty]$; the conventions for arithmetic and order in $[0,\infty]$ and for the sum $\sum_{m\in\mathbb{N}}a_{m}$ of a sequence in $[0,\infty]$ are those fixed in that definition. A subset $E$ of $\mathbb{R}^{q}$ is null if it is $\lambda_{q}$-null, that is, if $E\subseteq B$ for some $B\in\mathcal{B}(\mathbb{R}^{q})$ with $\lambda_{q}(B)=0$; such an $E$ need not itself belong to $\mathcal{B}(\mathbb{R}^{q})$. A property is said to hold almost everywhere when the set of points at which it fails is null, in the sense of A Property Holding Almost Everywhere. ¶

6. (Images, restrictions, and Lipschitz maps)¶ For a map $T$ with domain $D$ and a subset $A\subseteq D$ we write $T(A)=\{T(x):x\in A\}$ for the image of $A$ and $T|_{A}$ for the restriction of $T$ to $A$. For nonempty $E\subseteq\mathbb{R}^{q}$ and $p\in\mathbb{N}$ with $1\le p$, a map $T:E\to\mathbb{R}^{p}$ is called Lipschitz with constant $L$ always with respect to the Euclidean distances restricted to $E$ and to $\mathbb{R}^{p}$; explicitly, this means that $\lVert T(x)-T(y)\rVert\le L\lVert x-y\rVert$ for all $x,y\in E$.