Author:
(1) Attila Losonczi. Table of Links Abstract and 1 Introduction 1.1 Basic notions and notations 1.2 Basic definitions from [7] and [8] 2 Generalized integral 2.1 Multiplication on [0, +∞) × [−∞, +∞] 2.2 Measurability 2.3 The integral of functions taking values in [0, +∞) × [0, +∞) 3 Applications 4 References 2.1 Multiplication on [0, +∞) × [−∞, +∞] In order to define the integral, we will need multiplication on [0, +∞) × [0, +∞). We will use a natural way to do that, however one might argue against it, saying that that rule does not always hold for Hausdorff dimension and measure. We admit that it is true, but we could not find a better way until now, and more importantly, this method of multiplication and the resulting integral suit our later purposes completely. We define the operation on a greater domain that we will actually need. Remark 2.2. The multiplication of a real number and a value from [0, +∞) × [−∞, +∞] is a special case of the multiplication defined here, because for c ∈ R,(d, m) ∈ [0, +∞)×[−∞, +∞], we have c(d, m) = (0, c)(d, m) = (d, cm) if c 6= 0, while if c = 0, then both sides equal to (0, 0). Therefore if we identify c with (0, c), then we can embed the former algebraic structure into the current one. Proof. The sufficiency is obvious. Proposition 2.4. The multiplication is commutative and associative. Proof. Being commutative is trivial. Proposition 2.5. The distributive law holds in [0, +∞) × [0, +∞] Example 2.6. The distributive law does not hold in general if we allow negative numbers in the second coordinate as well. See e.g. (1, 1) (0, 5) + (0, −5)) = (0, 0) 6= (1, 5) + (1, −5) = (1, 0). This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Author: (1) Attila Losonczi. Author: Author: (1) Attila Losonczi. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 Basic notions and notations 1.1 Basic notions and notations 1.2 Basic definitions from [7] and [8] 1.2 Basic definitions from [7] and [8] 2 Generalized integral 2.1 Multiplication on [0, +∞) × [−∞, +∞] 2.1 Multiplication on [0, +∞) × [−∞, +∞] 2.2 Measurability 2.2 Measurability 2.3 The integral of functions taking values in [0, +∞) × [0, +∞) 2.3 The integral of functions taking values in [0, +∞) × [0, +∞) 3 Applications 3 Applications 4 References 4 References 2.1 Multiplication on [0, +∞) × [−∞, +∞] In order to define the integral, we will need multiplication on [0, +∞) × [0, +∞). We will use a natural way to do that, however one might argue against it, saying that that rule does not always hold for Hausdorff dimension and measure. We admit that it is true, but we could not find a better way until now, and more importantly, this method of multiplication and the resulting integral suit our later purposes completely. We define the operation on a greater domain that we will actually need. Remark 2.2. The multiplication of a real number and a value from [0, +∞) × [−∞, +∞] is a special case of the multiplication defined here, because for c ∈ R,(d, m) ∈ [0, +∞)×[−∞, +∞], we have c(d, m) = (0, c)(d, m) = (d, cm) if c 6= 0, while if c = 0, then both sides equal to (0, 0). Therefore if we identify c with (0, c), then we can embed the former algebraic structure into the current one. Remark 2.2. Proof . The sufficiency is obvious. Proof Proposition 2.4. The multiplication is commutative and associative. Proposition 2.4. The multiplication is commutative and associative. Proof . Being commutative is trivial. Proof Proposition 2.5. The distributive law holds in [0, +∞) × [0, +∞] Proposition 2.5. The distributive law holds in [0, +∞) × [0, +∞] Example 2.6. The distributive law does not hold in general if we allow negative numbers in the second coordinate as well. Example 2.6. The distributive law does not hold in general if we allow negative numbers in the second coordinate as well. See e.g. (1, 1) (0, 5) + (0, −5)) = (0, 0) 6= (1, 5) + (1, −5) = (1, 0). See e.g. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. available on arxiv

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

Generalized Hausdorff Integral and Its Applications: Multiplication on [0, +∞) × [−∞, +∞]

About Author

Comments

TOPICS

THIS ARTICLE WAS FEATURED IN

Related Stories

A Brief Review of Modern Shortest Path Algorithms and Graph Optimization Techniques

Generalized Hausdorff Integral and Its Applications: Abstract and Introduction

Generalized Hausdorff Integral and Its Applications: Basic Notions and Notations

Generalized Hausdorff Integral and Its Applications: Measurability

Generalized Hausdorff Integral and Its Applications: Integral of Functions

Generalized Hausdorff Integral and Its Applications: References

A Brief Review of Modern Shortest Path Algorithms and Graph Optimization Techniques

Generalized Hausdorff Integral and Its Applications: Abstract and Introduction

Generalized Hausdorff Integral and Its Applications: Basic Notions and Notations

Generalized Hausdorff Integral and Its Applications: Measurability

Generalized Hausdorff Integral and Its Applications: Integral of Functions

Generalized Hausdorff Integral and Its Applications: References

Light-Mode

Classic

Newspaper

Minty

Dark-Mode

Neon Noir

Minty

HN StartUps