An Expository Lecture of María Jesús Chasco on Some Applications of Fubini’s Theorem

The usefulness of Fubini’s theorem as a measurement instrument is clearly understood from its multiple applications in Analysis, Convex Geometry, Statistics or Number Theory. This article is an expository paper based on a master class given by the second author at the University of Vigo and is devoted to presenting some Applications of Fubini’s theorem. In the first part, we present Brunn–Minkowski’s and Isoperimetric inequalities. The second part is devoted to the estimations of volumes of sections of balls in Rn.

Fubini’s Theorem

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.

Lineability within Peano Curves, Martingales, and Integral Theory

This paper is devoted to give several improvements of some known facts in lineability approach. In particular, we prove that (i) the set of continuous mappings from the unit interval onto the unit square contains a closed,c-semigroupable convex subset, (ii) the set of pointwise convergent martingales(Xn)n∈NwithEXn→∞isc-lineable, (iii) the set of martingales converging in measure but not almost surely isc-lineable, (iv) the set of sequences(Xn)n∈Nof independent random variables, withEXn=0,∑n=1∞var Xn=∞, and the property that(X1+⋯+Xn)n∈Nis almost surely convergent, isc-lineable, (v) the set of bounded functionsf:[0,1]×[0,1]→Rfor which the assertion of Fubini’s Theorem does not hold is consistent withZFC 1-lineable (it is not 2-lineable), (vi) the set of unbounded functionsf:[0,1]×[0,1]→Rfor which the assertion of Fubini’s Theorem does not hold (with infinite integral allowed) isc-lineable but notc+-lineable.

Fubini’s Theorem for Non-Negative or Non-Positive Functions

Summary The goal of this article is to show Fubini’s theorem for non-negative or non-positive measurable functions [10], [2], [3], using the Mizar system [1], [9]. We formalized Fubini’s theorem in our previous article [5], but in that case we showed the Fubini’s theorem for measurable sets and it was not enough as the integral does not appear explicitly. On the other hand, the theorems obtained in this paper are more general and it can be easily extended to a general integrable function. Furthermore, it also can be easy to extend to functional space Lp [12]. It should be mentioned also that Hölzl and Heller [11] have introduced the Lebesgue integration theory in Isabelle/HOL and have proved Fubini’s theorem there.

Fubini’s Theorem on Measure

Summary The purpose of this article is to show Fubini’s theorem on measure [16], [4], [7], [15], [18]. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of all, for the product measure constructed in [14], we show some theorems. Then we introduce the section which plays an important role in Fubini’s theorem, and prove the relevant proposition. Finally we show Fubini’s theorem on measure.

Leibnizs rule and Fubinis theorem associated with Hahn difference operators

In $1945$, Wolfgang Hahn introduced his difference operator $D_{q,\omega}$, which is defined by where $\displaystyle{\omega_0=\frac {\omega}{1-q}}$ with $0<q<1, \omega>0.$ In this paper, we establish Leibniz's rule and Fubini's theorem associated with this Hahn difference operator.