Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background

(i) Base. This setting extends Measurable Functions and the Lebesgue Integral: Standing Notation and Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation: their conventions and notation, and through them those of Measurable Functions and the Lebesgue Integral: Notation and Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, are in force.

(ii) Product measures. Notation as in Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation (ii). Background: The Product σ-Algebra: Rectangles, Generators, Sections, Projections, Pairings and the Swap, Measures of Sections of a Product-Measurable Set, Existence and Uniqueness of the Product Measure, Tonelli's and Fubini's Theorems and Null Sets of a Product Measure, and Exchanging the Factors; for the formal bookkeeping: Rectangles, Sections and the Swap: Set Identities.

(iii) Image measures and densities. Notation as in Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation (iii). Background: Image Measures and Measures with a Density: Measures, Integrals and Change of Variables and Integrals of Finite Sums of Nonnegative Functions, and Constant Multiples of a Measure; for the formal bookkeeping: Precomposition Commutes with Pointwise Operations, and Sections of Functions Are Precompositions.

(iv) Lebesgue measure on $\mathbb{R}^{n}$. Notation as in Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation (iv). Background: Borel Sets of ℝ^{m+n} and the Product σ-Algebra of ℝ^m and ℝ^n, Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space, Existence and Uniqueness of Lebesgue Measure on ℝⁿ, Lebesgue Measure on ℝ^{m+n} as the Product of Lebesgue Measures on ℝ^m and ℝ^n, Lebesgue Measure on ℝⁿ: σ-Finiteness, Bounded Sets, Open Sets and Coordinate Hyperplanes, Translation and Reflection Invariance of Lebesgue Measure on ℝⁿ and Scaling Lebesgue Measure on ℝⁿ by a Diagonal Matrix; for the formal bookkeeping: Open Boxes: Emptiness, Intersections and Preimages under Concatenation.

(v) Linear change of variables. Background: Every Invertible Real Matrix Is an Orthogonal Matrix Times a Positive Diagonal Matrix Times an Orthogonal Matrix, Translation-Invariant Measures on ℝⁿ Are Multiples of Lebesgue Measure and Linear Change of Variables for Lebesgue Measure on ℝⁿ.