The Space of Metric Measure Spaces
Abstract
This thesis is broadly divided in two parts. In the first part we give a survey of various distances between metric spaces, namely the uniform distance, Lipschitz distance, Hausdor distance and the Gramoz Hausdor distance. Here we talk about only the most basic of their properties and give a few illustrative examples. As we wish to study collections of metric measure spaces, which are triples (X; d; m) consisting of a complete separable metric space (X; d) and a Boral probability measure m on X, there are discussions about some distances between them. Among the three that we discuss, the transportation and distortion distances were introduced by Sturm. The later, denoted by 2, on the space X2 of all metric measure spaces having finite L2size is the focus of the second part of this thesis.
The second part is an exposition based on the work done by Sturm. Here we prove a number of results on the analytic and geometric properties of (X2; 2). Beginning by noting that (X2; 2) is a noncomplete space, we try to understand its completion. Towards this end, the notion of a gauged measure space is useful. These are triples (X; f; m) where X is a Polish space, m a Boral probability measure on X and f a function, also called a gauge, on X X that is symmetric and square integral with respect to the product measure m2. We show that,
Theorem 1. The completion of (X2; 2) consists of all gauged measure spaces where the gauges satisfy triangle inequality almost everywhere. We denote the space of all gauged measure spaces by Y. The space X2 can be embedded in Y and the transportation distance 2 extends easily from X2 to Y. These two spaces turn out to have similar geometric properties.
On both these spaces 2 is a strictly intrinsic metric; i.e. any two members in them can be joined by a shortest path. But more importantly, using a description of the geodesics in these spaces, the following result is proved.
Theorem 2. Both (X2; 2) and (Y; 2) have nonnegative curvature in the sense of Alexandrov.
Collections
 Mathematics (MA) [148]
Related items
Showing items related by title, author, creator and subject.

Induction Motor Drives Based on Multilevel Dodecagonal and Octadecagonal Volatage Space Vectors
Mathew, K (20180321)For medium and highvoltage drive applications, multilevel inverters are very popular. It is due to their superior performance compared to 2level inverters such as reduced harmonic content in the output voltage and current, ... 
Investigations On Dodecagonal Space Vector Generation For Induction Motor Drives
Das, Anandarup (20110127)Multilevel converters are finding increased attention in industry and academia as the preferred choice of electronic power conversion for high power applications. They have a wide application area in a variety of industries ... 
NonLinear Finite Element Analysis Using StrainSpace Plasticity Coupled With Damage
Dawari, Balkrishna Maruti (20110127)The Thesis deals with StrainSpace Plasticity and its implementation in Nonlinear Finite Element framework coupled with damage. Conventional StressSpace Plasticity, though very popular amongst commercial nonlinear FEM ...