Extended Real-Valued Double Sequence and Its Convergence

Endou, Noboru

doi:10.1515/forma-2015-0021

Formalized Mathematics

2015

DOI: 10.1515/forma-2015-0021

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Extended Real-Valued Double Sequence and Its Convergence

Noboru Endou

Abstract: In this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.

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Cited by 2 publications

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Integral of Non Positive Functions

Endou

2017

Formalized Mathematics

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Summary.In this article, we formalize in the Mizar system [1,7] PreliminariesLet X be a non empty set and f be a non-negative partial function from X to R. Observe that −f is non-positive.Let f be a non-positive partial function from X to R. One can check that −f is non-negative. Now we state the propositions: (1) Let us consider a non empty set X, a non-positive partial function f from X to R, and a set E. Then f E is non-positive. (2) Let us consider a non empty set X, a set A, a real number r, and a partial function f from X to R. (ii) if c 0, then c · f is non-negative.

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Integral of Non Positive Functions

Endou

2017

Formalized Mathematics

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Absolutely Integrable Functions

Endou

2022

Formalized Mathematics

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Summary The goal of this article is to clarify the relationship between Riemann’s improper integrals and Lebesgue integrals. In previous articles [6], [7], we treated Riemann’s improper integrals [1], [11] and [4] on arbitrary intervals. Therefore, in this article, we will continue to clarify the relationship between improper integrals and Lebesgue integrals [8], using the Mizar [3], [2] formalism.

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Extended Real-Valued Double Sequence and Its Convergence

Abstract: In this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.

Cited by 2 publications

References 5 publications

Integral of Non Positive Functions

Integral of Non Positive Functions

Absolutely Integrable Functions

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