Atomicity through Fractal Measure Theory: Mathematical and Physical Fundamentals with Applications

Preface
Contents
1 Several hypertopologies: A short overview
	1.1 Introduction
	1.2 Vietoris and Hausdorff topologies
	1.3 Wijsman topology
	1.4 Several comparisons among the three topologies
	1.5 Kuratowski convergence
	References
2 A mathematical-physical approach on regularity in hit-and-miss hypertopologies for fuzzy set multifunctions
	2.1 Introduction
	2.2 Hit-and-miss hypertopologies. An overview
		2.2.1 Vietoris topology
		2.2.2 Wijsman topology
		2.2.3 Hausdorff topology
	2.3 Regular set multifunctions
	References
3 Non-atomic set multifunctions
	3.1 Basic notions, terminology and results
	3.2 Non-atomicity for set multifunctions
	3.3 An extension by preserving non-atomicity
	References
4 Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions
	4.1 Introduction
	4.2 Preliminary definitions and remarks
	4.3 Non-atomic multisubmeasures
	4.4 The Darboux property
		4.4.1 The Darboux property for multisubmeasures
		4.4.2 The Darboux property for multimeasures
	4.5 Conclusions
	References
5 Atoms and pseudo-atoms for set multifunctions
	5.1 Basic notions, terminology and results
	5.2 Pseudo-atoms for set multifunctions
	5.3 Darboux property for set multifunctions
	References
6 Gould integrability on atoms for set multifunctions
	6.1 Basic notions, terminology and results
	6.2 Measurability on atoms
	6.3 Gould integrability on atoms
	References
7 Continuity properties and Alexandroff theoremin Vietoris topology
	7.1 Introduction
	7.2 Preliminaries
	7.3 Continuity properties in Vietoris topology
	7.4 Regularity and Alexandroff theorem in Vietoris topology
	7.5 Conclusions
	References
8 Approximation theorems for fuzzy set multifunctions in Vietoris topology: Physical implications of regularity
	8.1 Introduction
	8.2 Terminology, basic notions and results
	8.3 Convergences for real-valued measurable functions with respect to P0(X)-valued monotone set multifunctions in Vietoris topology
	8.4 Set-valued Egoroff and Lusin type theorems and applications in the Vietoris topology
	8.5 Regularization by sets of functions of -approximation typescale
	8.6 Physical implications
	8.7 Conclusions
	References
9 Atomicity via regularity for non-additive set multifunctions
	9.1 Basic notions, terminology and results
	9.2 Non-atomicity via regularity
	References
10 Extended atomicity through non-differentiability and its physical implications
	10.1 Introduction
	10.2 Towards Quantum Measure Theory by means of Fractal Mechanics
	10.3 Types of atoms in the mathematical approach
		10.3.1 Atoms and pseudo-atoms
		10.3.2 Minimal atoms
		10.3.3 Coherence and decoherence through Young type experiments
	10.4 Fractal Mechanics and some applications
		10.4.1 Fractal operator and its implications
		10.4.2 Stationary dynamics of a complex system structural units in the fractal Schrödingerrepresentation
	10.5 From the standard mathematical atom to the fractal atom by means of a physical procedure
	References
11 On a multifractal theory of motion in a non-differentiable space: Toward a possible multifractal theory of measure
	11.1 Introduction
	11.2 Consequences of non-differentiability on a space manifold
		11.2.1 Fractal fluid geodesics
	11.3 Fractality and its implications
	11.4 Fractal geodesics in the Schrödinger type representation
	11.5 The one dimensional potential barrier: Fractal tunnel type effect
	11.6 Fractal motions in central field
	11.7 Complex system geodesics in the fractal hydrodynamic representation
	References
List of symbols
Index