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 2.2 Measurability First we need to define measurability of generalized functions. Definition 2.10. Let (K, S) be a measurable space (i.e. S is a σalgebra on P(K)). Let f : K → [0, +∞) × [0, +∞) be a function. We say that f is measurable if for each (d, m) ∈ [0, +∞) × [0, +∞) {x ∈ K : f(x) < (d, m)} ∈ S holds. Proposition 2.11. Let (K, S) be a measurable space and f : K → [0, +∞) × [0, +∞). Then the following statements are equivalent. f is measurable.


For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : f(x) ≤ (d, m)} ∈ S.


For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) < f(x)} ∈ S.


For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) ≤ f(x)} ∈ S. Proof. All equivalences simply follow from the fact that the space [0, +∞) × [0, +∞] is first countable and T2. Proposition 2.12. Let (K, S) be a measurable space and f : K → [0, +∞) × [0, +∞) be measurable. Then the following statements hold. Let K = [0, 1], S be the Borel sets, g: [0, 1] → [0, 1] be a non Borel measurable function and f(x) = (x, g(x)) when x ∈ [0, 1]. Clearly {x ∈ K: f(x) < (d, m)} equals to either {x ∈ K: x < d} or {x ∈ K: x ≤ d}, and both sets are Borel, hence f is measurable. Proposition 2.14. Let (K, S) be a measurable space and f: K → [0, +∞) × [0, +∞) be measurable and (d′, m′) ∈ [0, +∞)×[0, +∞). Then (d′, m′)f is measurable as well. Proof. Let (d, m) ∈ [0, +∞) × [0, +∞). If (d ′, m′) = (0, 0), then the statement is trivial. If m′ = 0, d′ > 0, then {x: (d′, m′)f(x) ≤ (d, m)} = {x: f(x) ≤ (d − d′, +∞)}, where similar argument works Proposition 2.15. Let (K, S) be a measurable space and f, g : K → [0, +∞) × [0, +∞) be measurable. Then f + g is measurable as well. 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 2.2 Measurability First we need to define measurability of generalized functions. Definition 2.10. Let (K, S) be a measurable space (i.e. S is a σalgebra on P(K)). Let f : K → [0, +∞) × [0, +∞) be a function. We say that f is measurable if for each (d, m) ∈ [0, +∞) × [0, +∞) {x ∈ K : f(x) < (d, m)} ∈ S holds. Definition 2.10. measurable Proposition 2.11. Let (K, S) be a measurable space and f : K → [0, +∞) × [0, +∞). Then the following statements are equivalent. Proposition 2.11. Let (K, S) be a measurable space and f : K → [0, +∞) × [0, +∞). Then the following statements are equivalent. f is measurable. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : f(x) ≤ (d, m)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) < f(x)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) ≤ f(x)} ∈ S. f is measurable. f is measurable. f is measurable. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : f(x) ≤ (d, m)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : f(x) ≤ (d, m)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : f(x) ≤ (d, m)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) < f(x)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) < f(x)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) < f(x)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) ≤ f(x)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) ≤ f(x)} ∈ S. For each (d, m) ∈ [0, +∞) × [0, +∞], {x ∈ K : (d, m) ≤ f(x)} ∈ S. Proof . All equivalences simply follow from the fact that the space [0, +∞) × [0, +∞] is first countable and T2. Proof Proposition 2.12. Let (K, S) be a measurable space and f : K → [0, +∞) × [0, +∞) be measurable. Then the following statements hold. Proposition 2.12. Let (K, S) be a measurable space and f : K → [0, +∞) × [0, +∞) be measurable. Then the following statements hold. Let K = [0, 1], S be the Borel sets, g: [0, 1] → [0, 1] be a non Borel measurable function and f(x) = (x, g(x)) when x ∈ [0, 1]. Clearly {x ∈ K: f(x) < (d, m)} equals to either {x ∈ K: x < d} or {x ∈ K: x ≤ d}, and both sets are Borel, hence f is measurable. Let K = [0, 1], S be the Borel sets, g: [0, 1] → [0, 1] be a non Borel measurable function and f(x) = (x, g(x)) when x ∈ [0, 1]. Clearly {x ∈ K: f(x) < (d, m)} equals to either {x ∈ K: x < d} or {x ∈ K: x ≤ d}, and both sets are Borel, hence f is measurable. Proposition 2.14. Let (K, S) be a measurable space and f: K → [0, +∞) × [0, +∞) be measurable and (d′, m′) ∈ [0, +∞)×[0, +∞). Then (d′, m′)f is measurable as well. Proposition 2.14. Let (K, S) be a measurable space and f: K → [0, +∞) × [0, +∞) be measurable and (d′, m′) ∈ [0, +∞)×[0, +∞). Then (d′, m′)f is measurable as well. Proof . Let (d, m) ∈ [0, +∞) × [0, +∞). Proof If (d ′, m′) = (0, 0), then the statement is trivial. If m′ = 0, d′ > 0, then {x: (d′, m′)f(x) ≤ (d, m)} = {x: f(x) ≤ (d − d′, +∞)}, where similar argument works Proposition 2.15. Let (K, S) be a measurable space and f, g : K → [0, +∞) × [0, +∞) be measurable. Then f + g is measurable as well. Proposition 2.15. Let (K, S) be a measurable space and f, g : K → [0, +∞) × [0, +∞) be measurable. Then f + g is measurable as well. 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: Measurability

About Author

Comments

TOPICS

THIS ARTICLE WAS FEATURED IN

Related Stories

Untitled Story

Generalized Hausdorff Integral and Its Applications: Abstract and Introduction

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: Multiplication on [0, +∞) × [−∞, +∞]

Generalized Hausdorff Integral and Its Applications: Integral of Functions

Generalized Hausdorff Integral and Its Applications: Abstract and Introduction

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: Multiplication on [0, +∞) × [−∞, +∞]

Generalized Hausdorff Integral and Its Applications: Integral of Functions

Light-Mode

Classic

Newspaper

Dark-Mode

Neon Noir

Minty

HN StartUps