xiv Notation measure μ, with the usual identification of functions which coincide almost everywhere. Whenever p 1, we shall denote by p its conjugate exponent: 1 p + 1 p = 1. Whenever T is a map from a measure space X, equipped with a measure μ, to an arbitrary space Y , we denote by T the image measure (or push- forward) of μ by T . Explicitly, (T #μ)[B] = μ[T −1(B)], where T −1(B) = {x X T (x) B}. The set of all T : X X such that T = μ will be denoted by S(X). We shall always use push-forward in this sense: when we write T #f = g, where f and g are nonnegative functions, this means that the measure having density f is pushed foward to the measure having density g (usually the reference measure will be the Lebesgue measure). If X is a topological space, then it will be equipped with its Borel σ- algebra. We shall denote by C(X) the space of continuous functions on X by Cb(X) the space of bounded continuous functions on X and by C0(X) the space of continuous functions on X going to 0 at infinity. Sometimes these notations will be replaced by C(X R), Cb(X R), C0(X R). The space Cb(X) comes with a natural norm, u = supX |u|. Whenever A X, we denote by Int(A) the largest open set contained in A, and by A the smallest closed set containing A. We set ∂A = A \ Int(A). By definition, the support of a measure μ on X will be the smallest closed set F X with μ[X \ F ] = 0, and will be denoted Supp μ. On the other hand, when we say that μ is concentrated on A X, this only means that μ[X \ A] = 0, without A being necessarily closed. If X is a metric space, we shall equip it with the topology induced by its distance, and denote by B(x, r) the ball of radius r and center x. We shall denote by Lip(X) the set of all Lipschitz functions on X we shall also denote by Pp(X) the space of Borel probability measures μ on X with finite moment of order p, in the sense that d(x0,x)p dμ(x) +∞ for some (and thus any) x0 X. When X is a Banach space and X∗ its topological dual, we shall denote by ·, · the duality bracket between X and X∗. A particular case of this is the scalar product in a Hilbert space. If ϕ is a convex function on a Banach space X, then ϕ∗ will stand for its dual convex function, in the sense of Legendre-Fenchel duality. The subdifferential of ϕ will be denoted by ∂ϕ, and identified with its graph, which is a subset of X × X∗. Basic definitions for these objects are recalled
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