In this chapter, directly inspired from . we define almost all notions used throughout this book: traffic plan, its irrigated and irrigating measure, its transference plan, its parameterization, its fibers, the stopping time of each fiber and the multiplicity (or flow) at each point. After addressing some measurability issues we prove semicontinuity properties for all quantities of interest (average length of fibers, stopping time and multiplicity). All of these notions are first applied to the classical Monge-Kantorovich existence theory of optimal transports. Finally an existence theorem is given for traffic plans with prescribed irrigating and irrigated measures (the irrigation problem) and with prescribed transference plan (the who goes where problem).
KeywordsLower Semicontinuity Measurable Subset Optimal Transport Lower Semicontinuous Function Monotone Convergence Theorem
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