Fractional Probability Measure and Its Properties

Abstract

Based on recent studies by Guy Jumarie [1] which defines probability density of fractional order and fractional moments by using fractional calculus (fractional derivatives and fractional integration), this study expands the concept of probability density of fractional order by defining the fractional probability measure, which leads to a fractional probability theory parallel to the classical one. According to the probability principles in classical probability theory and the definition of probability density of fractional order by Guy Jumarie, at first, the fractional probability principles are discussed. Then the fractional probability space is introduced. Consequently, the fractional probability measure , is explained. Moreover, validity of the classical "probability measure continuity" theorem ( ) for the fractional probability measure is verified, which results in "Fatou Lemma" and some theorems in convergence concept.