Besov regularity of stochastic measures

Let μμ be a random σσ-additive in probability set function defined on Borel subsets of [a,b][a,b]. We prove that if the process μ([a,t]),a⩽t⩽b, has continuous paths, then they belong a.s. to the Besov space Bppα([a,b]) for all p⩾2,0<α<1/p.