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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.

We consider a Köthe space $(,|\left|·\right|{\left|}_{\right)}$ of random variables (r.v.) defined on the Lebesgue space ([0,1],B,λ). We show that for any sub-σ-algebra ℱ of B and for all r.v.’s X with values in a separable finitely compact metric space (M,d) such that d(X,x) ∈ for all x ∈ M (we then write X ∈ (M)), there exists a median of X given ℱ, i.e., an ℱ-measurable r.v. Y ∈ (M) such that ${\left|\right|d(X,Y)\left|\right|}_{\le }\left|\right|d(X,Z)\left|\right|$ for all ℱ-measurable Z. We develop the basic theory of these medians, we show the convergence of empirical medians and we give some applications....