By R. Switzer
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Extra info for Algebraic Topology, Homotopy and Homology
12, establishes this Riemann integral representation continuously over the base space, in the context of an aﬃne Rq -bundle. 11. Let X ∈ Rq , x ∈ X, and let z ∈ IntConv(X, x). Each Cstructure (g, G) ∈ int Xxz can be reparametrized to a C-structure (h, H) such that 1 z = 0 h(t) dt. Proof. Let (g, G) ∈ int Xxz be a C-structure and let 0 = s0 < s1 · · · < sn+1 = 1 be a partition of the interval [0, 1] such that z is contained in the interior of the convex hull of the points g(si ), 1 ≤ i ≤ n. For each i, 1 ≤ i ≤ n, let dµi be a 1 positive measure on [0, 1] such that 0 dµi = 1, and dµi ≈ δ(s − si ).
E. the parameter space C is a point, this problem, with a diﬀerent analytic formulation, occurs in Optimal Control theory and is solved by the Relaxation Theorem (cf. A. F. Filippov , and Chapter X below for a discussion of the Relaxation Theorem). 4 may be viewed as a C 1 -Relaxation Theorem with parameters, and applies to relations R which are open in E. On the other hand, §1. THE ONE-DIMENSIONAL THEOREM 35 in case B = [0, 1] and under Lipschitz hypotheses on a closed relation R, the Relaxation Theorem of Filippov  provides a section f ∈ Γ(E) for which the C 0 -approximation (ii) holds and such that the section ∂t f : B → R is absolutely continuous in t (not C 1 in t in general).
9) that, 1 1 h(s) ds = 0 1 g ◦ λ−1 (s) ds = 0 g dµ = z. 11) 0 Let H : [0, 1]2 → X be the reparametrization, H(t, s) = G(λ−1 (t), s). 11) obtains. 12 (Integral Representation). Let p : E → B be an aﬃne Rq -bundle over a second-countable paracompact space B. Let R ⊂ E be open and suppose β ∈ Γ(R) and f ∈ Γ(E) satisfy the property that, for all b ∈ B, f (b) ∈ Conv(Rb , β(b)). Each C-structure (g, G) over B with respect to f, β can be reparametrized to a C-structure (h, H) such that for all b ∈ B (recall h : [0, 1] → Γ(R)), f (b) = 1 h(t, b) dt.
Algebraic Topology, Homotopy and Homology by R. Switzer