A characterization of the transport property is given. New properties for strongly nonatomic probabilities are established. We study the relationship between the nondifferentiability of a real function f and the fact that the probability measure ${\lambda }_{f*}:=\lambda ◦{(f*)}^{-1}$, where f*(x):=(x,f(x)) and λ is the Lebesgue measure, has the transport property.

Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set $A$ we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s...

The existence of a minimal element in every equivalence class of pairs of bounded closed convex sets in a reflexive locally convex topological vector space is proved. An example of a non-reflexive Banach space with an equivalence class containing no minimal element is presented.