On a Class of Universal Probability Spaces: Case of Complex Fields

Abstract

The objective of this paper is to extend the Universal Probability Space (UPS) in [1] to include complex events. The UPS consists of Borel sets, elements of which are tensors. It is shown that the UPS has a defined metric and this metric is in fact the probability measure (P). The metric as a probability measure is proven to exist for any tensor event (x ∈ Rd) in the space of all tensor fields, (Rd). In this paper it is shown that for any complex event, (x ∈ Cd) in a space of all complex tensor fields, (Cd), a probability measure (P) in the form of a metric exists. To this effect several theorems are introduced and proven, mainly by modifying concepts introduced in [2], [3], [4], [5], to include complex fields. Finally following [6], [7], [8], a case is demonstrated in order to compare probability as a metric for complex events with classical probability. The objective of the case study is to show that metric probability is a more realistic measure than classical probability for complex events.