Lebesgue Integration on Rn

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.

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Shape analysis, Lebesgue integration and absolute continuity connections

10.6028/nist.ir.8217 ◽

2018 ◽

Author(s):

Javier Bernal

Keyword(s):

Shape Analysis ◽

Absolute Continuity ◽

Lebesgue Integration

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The Convergence Theorems of Lebesgue Integration Theory

Universitext - Postmodern Analysis ◽

10.1007/978-3-662-03635-8_17 ◽

1998 ◽

pp. 199-208

Author(s):

Jürgen Jost

Keyword(s):

Integration Theory ◽

Convergence Theorems ◽

Lebesgue Integration

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Information Measures

Formalized Probability Theory and Applications Using Theorem Proving ◽

10.4018/978-1-4666-8315-0.ch009 ◽

2015 ◽

pp. 129-142

Keyword(s):

Information Theory ◽

Mutual Information ◽

Order Logic ◽

Unified Framework ◽

Higher Order Logic ◽

Absolutely Continuous ◽

Information Measures ◽

Finite Spaces ◽

Nikodym Derivative ◽

Lebesgue Integration

This chapter presents a higher-order-logic formalization of the main concepts of information theory (Cover & Thomas, 1991), such as the Shannon entropy and mutual information, using the formalization of the foundational theories of measure, Lebesgue integration, and probability. The main results of the chapter include the formalizations of the Radon-Nikodym derivative and the Kullback-Leibler (KL) divergence (Coble, 2010). The latter provides a unified framework based on which most of the commonly used measures of information can be defined. The chapter then provides the general definitions that are valid for both discrete and continuous cases and then proves the corresponding reduced expressions where the measures considered are absolutely continuous over finite spaces.

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