Convex analysis by R. Tyrrell Rockafellar

By R. Tyrrell Rockafellar

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Let ωi ∈ Sp,j , ωi → ω. Then, for each of the curves χ(ωi ), the sum of the lengths of the loops of length ≥ p1 is ≥ λ − tm . Letting i → ∞, we deduce that the sum of the lengths of the loops of χ(ω) of length ≥ p1 is also ≥ λ − tm . In other words, ω ∈ Sp,j . Since Sp = ∪j Sp,j ∪ N where N is a null set, we deduce that Sp is a measurable set. Now, since c = Tωc , we have that ∪p Tω,p c ∩ [0, λ]| ≥ λ − tm } {ω : |Tωc ∩ [0, λ]| ≥ λ − tm } = {ω : sup |Tω,p p = j k 1 c {ω : |Tω,k ∩ [0, λ]| ≥ λ − tm − }.

12. The measurability of Tχ : Ω → R+ is an easy check. Indeed, {ω : Tχ (ω) ≤ t} = {ω : χ(ω, t ) = χ(ω, t )} t ,t ∈Q, t ,t ≥t which is a countable intersection of measurable subsets of Ω. 13. Let Ω be a measurable subset of R with finite measure. We call parameterized traffic plan a measurable map χ : Ω × R+ → X such that t → χ(ω, t) is 1-Lipschitz for all ω ∈ Ω and Ω Tχ (ω)dω < +∞. Without risk of ambiguity we shall call fiber both a path χ(ω, ·) and the range in RN of χ(ω, ·). Denote by |χ| := |Ω| the total mass transported by χ and by Pχ the law of ω → χ(ω) ∈ K defined by Pχ (E) = |χ−1 (E)| for every Borel set E ⊂ K.

10. Let P be a traffic plan with finite energy and χ a parameterization of P . Assume that almost all fibers have positive length. Then Sχ is countably rectifiable. , for x ∈ R \ ∪∞ j=1 Im χ(ωj ). 6) Proof. 8 implies that the set Sχ of points x such that |x|χ > 0 has σ−finite H1 -measure. Thus, by a classical result of Besicovich (see [39]), Sχ will be countably rectifiable if its H1 -density superior to 1 for almost every point in Sχ . We now prove that it is the case. 33 of the energy we know that there is Ω ⊂ Ω with |Ω | = 1 such that for every ω ∈ Ω the multiplicity of χ(ω, t) is positive at almost every point of [0, Tχ (ω)].

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