Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation

(i) Base. This setting extends Measurable Functions and the Lebesgue Integral: Notation and Euclidean Space and Real Matrices: Standing Notation: their conventions and notation are in force. Below, $(Y,\mathcal{G},\nu)$ is a further measure space and $(Z,\mathcal{H})$ is a measurable space; as in the extended settings, the notation is introduced for all of them at once, and a result adopting this setting uses it for whichever it names.

(ii) Products and sections. $\mathrm{Rect}(\mathcal{F},\mathcal{G})$ is the set of measurable rectangles, and $\mathcal{F}\otimes\mathcal{G}$ is the product $\sigma$-algebra on $X\times Y$. For $E\subseteq X\times Y$, $x\in X$ and $y\in Y$, $E_{x}\subseteq Y$ and $E^{y}\subseteq X$ are the sections of $E$; for a function $f$ on $X\times Y$, $f_{x}$ and $f^{y}$ are its sections. When $\mu$ and $\nu$ are $\sigma$-finite, $\mu\otimes\nu$ is their product measure on $(X\times Y,\mathcal{F}\otimes\mathcal{G})$. Background: The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap, Measures of Sections of a Product-Measurable Set and Existence and Uniqueness of the Product Measure.

(iii) Image measures and densities. For an $\mathcal{F}/\mathcal{H}$-measurable function $T:X\to Z$, $\mu\circ T^{-1}$ is the image measure of $\mu$ under $T$. For $h\in\mathcal{M}^{+}(X,\mathcal{F})$, $h\mu$ is the measure with density $h$ with respect to $\mu$. Background: Image Measures and Measures with a Density: Measures, Integrals and Change of Variables.

(iv) Lebesgue measure on $\mathbb{R}^{n}$ and images of sets. $\lambda_{n}:\mathcal{B}(\mathbb{R}^{n})\to[0,\infty]$ is Lebesgue measure on $\mathbb{R}^{n}$. For $a,b\in\mathbb{R}^{n}$, $R(a,b)$ is the open box with corners $a$ and $b$, and $\mathcal{R}_{n}$ is the set of open boxes in $\mathbb{R}^{n}$. $\tau:\mathbb{R}\to\mathbb{R}^{1}$ is the function sending $t$ to the $1$-tuple with component $t$, as in Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n (iii). For sets $D,E$, a function $T:D\to E$ and $B\subseteq D$, $T[B]$ is the image of $B$ under $T$; in particular, for $B\subseteq\mathbb{R}^{n}$, $a\in\mathbb{R}^{n}$, $c\in\mathbb{R}$ and $A\in\mathcal{M}_{p\times n}(\mathbb{R})$, the sets $B+a$, $-B$, $cB$ and $A[B]$ are the images of $B$ under the functions $x\mapsto x+a$, $x\mapsto-x$, $x\mapsto cx$ and $x\mapsto Ax$. Background: Existence and Uniqueness of Lebesgue Measure on ℝⁿ, Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n and Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space.