Fubini and Change of Variables

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1 · Prerequisites

Abstract Algebra

Combinatorics

Finite Counting, Factorials and Binomial Coefficients

Coxeter Groups

Symmetric Groups, Cycle Decomposition and the Sign Homomorphism

Linear Algebra

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Definition

A continuous function on a closed rectangle has repeated Riemann integrals in every coordinate order, all equal to its multiple integral

Corollary

A region between two continuous graphs is Jordan measurable, and a continuous integrand extending to its closure integrates by vertical sections

Definition

The Jacobian determinant of a square-dimensional C¹ map is the determinant of its Jacobian matrix

Lemma

Finite Jordan covers bound upper integrals, while interior-disjoint Jordan subfamilies bound lower integrals

Theorem

A linear endomorphism of ℝⁿ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant

Corollary

The Jordan content of the parallelepiped spanned by the columns of a square real matrix is the absolute value of its determinant

Lemma

A C¹ map uniformly close to the identity derivative sandwiches a cube between contracted and expanded cubes

Lemma

On a small cube, a C¹ diffeomorphism distorts Jordan content by factors arbitrarily close to its linearized absolute determinant

Theorem

An injective C¹ map with invertible derivative sends compact Jordan sets to compact Jordan sets

Lemma

A compact subset of an open Euclidean set has a compact Jordan neighborhood inside that open set

Lemma

A bounded open Jordan set has an increasing exhaustion by compact finite unions of grid rectangles with vanishing content remainder

Theorem

A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage

Corollary

Change of variables on bounded open Jordan sets when both integrands are bounded and Riemann integrable

Corollary

In one dimension the compact-Jordan formula is substitution over the unoriented image interval with the absolute derivative

Sources

Fubini and Change of Variables

3 · Items

Sections, lower and upper section integrals, and iterated Riemann integrals on product rectangles and Jordan sets

A product grid bounds the Darboux sums of the lower and upper section-integral functions

Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections

A continuous function on a closed rectangle has repeated Riemann integrals in every coordinate order, all equal to its multiple integral

The integral of a product function on a product rectangle is the product of the two integrals

An integrable function whose sections vanish outside finite sets has multiple integral zero

Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable

Cavalieri: Jordan content is the integral of sectional contents, and equal sections give equal content

A region between two continuous graphs is Jordan measurable, and a continuous integrand extending to its closure integrates by vertical sections

The Jacobian determinant of a square-dimensional C¹ map is the determinant of its Jacobian matrix

Finite Jordan covers bound upper integrals, while interior-disjoint Jordan subfamilies bound lower integrals

A linear endomorphism of ℝⁿ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant

The Jordan content of the parallelepiped spanned by the columns of a square real matrix is the absolute value of its determinant

A C¹ map uniformly close to the identity derivative sandwiches a cube between contracted and expanded cubes

On a small cube, a C¹ diffeomorphism distorts Jordan content by factors arbitrarily close to its linearized absolute determinant

An injective C¹ map with invertible derivative sends compact Jordan sets to compact Jordan sets

A compact subset of an open Euclidean set has a compact Jordan neighborhood inside that open set

A bounded open Jordan set has an increasing exhaustion by compact finite unions of grid rectangles with vanishing content remainder

Change of variables for an injective C¹ map on a compact Jordan set

The content of a compact Jordan image is the integral of the absolute Jacobian determinant

The support of a function on ℝⁿ and its compactly supported Riemann integral

The Riemann integral of a compactly supported function is independent of its bounding rectangle

A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage

Change of variables on bounded open Jordan sets when both integrands are bounded and Riemann integrable

In one dimension the compact-Jordan formula is substitution over the unoriented image interval with the absolute derivative