You have used a particular construction of B.M. that happens to admit a differentiable function (on a set of measure zero). But other constructions need not ha

You have used a particular construction of B.M. that happens to admit a differentiable function (on a set of measure zero). But other constructions need not have this property; in fact, you could make a version (in the technical sense of that term, see http://almostsure.wordpress.com/2009/11/03/stochastic-processes-indistinguishability-and-modifications/ http://almostsure.wordpress.com/2009/11/03/stochastic-proces...) of the B.M. that uses your construction, but removes the null set of differentiable sample paths. In other words, this property ("differentiability on null sets") is not characteristic of a B.M., it is incidental to your construction. I can't believe you bothered to fuss about these hypothetical properties on null sets when we are dealing with cadlag processes on a separable index set. Sheesh.