I read here that the Lebesgue differentiation theorem may be generalized to Borel measures which are locally doubling, i.e., where the measure of a Ball is bounded my some constant (possibly depending on the location of the center of the ball) times the measure of the same ball but with half the radius.
Is the Wiener measure on the measure space $C[0,T]$ of continuous paths (which is a separable metric space) with the Borel sigma algebra (using the sup norm topology) locally doubling?
If so would the zero path be a Lebesgue point in general? Or would that depend on the measurable function at hand?
