Generalized Hausdorff Integral and Its Applications: Basic Notions and Notations

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 1.1 Basic notions and notations Here we enumerate the basics that we will apply throughout the paper. For K ⊂ R, χK will denote the characteristic function of K i.e. χK(x) = 1 if x ∈ K, otherwise χK(x) = 0. Some usual operations and relation with ±∞: (+∞)+(+∞) = +∞, (−∞)+ (−∞) = −∞; if r ∈ R, then r+(+∞) = +∞, r+(−∞) = −∞, −∞ 0, then r · (+∞) = +∞, r · (−∞) = −∞. If r < 0, then r · (+∞) = −∞, r · (−∞) = +∞. And 0 · ∞ = 0. λ will denote the Lebesgue-measure and also the outer measure as well. We call f : K → R a simple function, if Ranf is finite. 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 1.1 Basic notions and notations Here we enumerate the basics that we will apply throughout the paper. For K ⊂ R, χK will denote the characteristic function of K i.e. χK(x) = 1 if x ∈ K, otherwise χK(x) = 0. Some usual operations and relation with ±∞: (+∞)+(+∞) = +∞, (−∞)+ (−∞) = −∞; if r ∈ R, then r+(+∞) = +∞, r+(−∞) = −∞, −∞ 0, then r · (+∞) = +∞, r · (−∞) = −∞. If r < 0, then r · (+∞) = −∞, r · (−∞) = +∞. And 0 · ∞ = 0. λ will denote the Lebesgue-measure and also the outer measure as well. We call f : K → R a simple function, if Ranf is finite. 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