On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation

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An optimal transport approach for seismic tomography: application to 3D full waveform inversion (2016, background): “In addition, when (14) admits a solution T , the measure π = (I, T )#µ is a solution of the relaxed problem (13) [Ambrosio, 2003, Pratelli, 2007].”
A User’s Guide to Optimal Transport (2013, unknown): “Pratelli [68].”
Adapted Wasserstein distances and stability in mathematical finance (2019, unknown): “[56].”
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below (2014, unknown): “For (ii) assume for a moment that μ has no atoms and recall that by a result of Pratelli ([47]) in this case for every ε > 0 there is a Borel map T : M2 → M2 such that T∗μ = ν and √∫ d1(x, T (x))2 dμ(x) ≤ W2(μ, ν) + ε.”
Chapter 1 – Gradient Flows of Probability Measures (2007, background): “This result can also be extended to classes of unbounded cost functions, see [79].”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, methodology): “The references on this subject are [253] and [8].”