Parametric curves in random metric spaces

This thesis consists of five chapters. In the first chapter, the historical developments of the theories of Metric Geometry and Random Metric (RM) Spaces are presented. In the second chapter, some basic concepts such as path (parametric curve), arc, length structure and length space, and results related to these concepts in Metric Geometry are presented. In the third chapter; the concepts of (ε,λ)-topology related to RM spaces, random normed space, path (parametric curve) in an RM space, random total variation, random length, almost rectifiability and non-rectifiability are presented. The fourth chapter is the original part of the thesis, and in this chapter; some basic results related to the random lengths and random diameters of paths (parametric curves) in RM spaces have been obtained. Furthermore, some properties of the spaces of these paths have been investigated. The fifth chapter is the conclusion part of the thesis; and in this chapter, the thesis is summarized.