Fubini’s Theorem

Abstract

Summary Fubini theorem is an essential tool for the analysis of high-dimensional space [8], [2], [3], a theorem about the multiple integral and iterated integral. The author has been working on formalizing Fubini’s theorem over the past few years [4], [6] in the Mizar system [7], [1]. As a result, Fubini’s theorem (30) was proved in complete form by this article.

A Coq Formalization of Lebesgue Induction Principle and Tonelli’s Theorem

“…Some Fubini-like results are available in HOL Light [13]. More recently, Tonelli's theorem was formalized in Mizar by Endou [11]. The formalizations nearest to ours are in Isabelle/HOL and Lean.…”

“…And the iterated integral is the integral (in X 1 ) of the integral (in X 2 ) of the sections of functions. Finally, Formula (10) is proved, and then (11) is deduced from the latter by a swap of variables relying both on a change of measure and on the uniqueness of the product measure. The main argument for this proof is the Lebesgue induction principle (see Section 3).…”

“…The first formula of Tonelli's theorem (10) applied to X 2 × X 1 gives (c), and Equation ( 12) gives (d). With swap f denoting f • h, the second part of Tonelli's theorem (11) is Statement of Tonelli's Theorem. Finally, assuming that X 1 and X 2 are nonempty and that µ 1 and µ 2 are σ-finite measures, we have (a more comprehensive theorem legitimating of all integrals is also provided as Theorem Tonelli):…”

“…A recent work in Coq has been developed for probability theory. 11 Many definitions are similar to ours, but in a simpler setting where measures are finite. Fubini's theorem also appeared in math-comp/analysis 12 after the submission of this article.…”