A simplified proof of the Daniell integral extension theorem in ordered spaces
Given a two-dimensional fractional multiplicative process determined by two Hurst exponents and , we show that there is an associated uniform Hausdorff dimension result for the images of subsets of by if and only if .
We characterize the autonomous, divergence-free vector fields on the plane such that the Cauchy problem for the continuity equation admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential associated to . As a corollary we obtain uniqueness under the assumption that the curl of is a measure. This result can be extended to certain non-autonomous vector fields with bounded divergence....