Embedding theorem for metric space with convex combination operation and applications
Nguyen Tran Thuan · arXiv (Cornell University) · 2015
In this paper, we embed metric space endowed with a convex combination operation into a Banach space and the embedding preserves the structures of metric and convex combination. For applications, on the one hand, some nice properties of expectation such as representation of expected value through continuous affine mappings, the linearity of expectation will be given. On the other hand, the notion of conditional expectation will be also introduced and discussed. Thanks to embedding theorem, we establish some basic properties of conditional expectation, Jensen's inequality, convergences of martingales and ergodic theorem.