Approach for a metric space with a convex combination operation and applications

In this study, we embed a metric space endowed with a convex combination operation, which is called a convex combination space, into a Banach space, where the embedding preserves the structures of the metric and convex combination. We also establish applications of this embedding for a random element that takes values in this type of space. On the one hand, we show some useful properties of mathematical expectation, such as the representation of expectation through continuous affine mappings and the linearity of expectation. On the other hand, the notion of conditional expectation is also introduced and discussed. Using this embedding theorem, we establish some basic properties of conditional expectation, Jensen's inequality, the convergences of martingales, and ergodic theorem.