On a Class of Universal Probability Spaces: Case of Complex Fields
Mahkame Khoshyaran · Advances in Research · 2016
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 ∈ R d ) in the space of all tensor fields, (R d ).In this paper it is shown that for any complex event, (x ∈ C d ) in a space of all complex tensor fields, (C d ), 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.