A Set of Axioms for Differential Geometry by Veblen O., Whitehead J. H.

By Veblen O., Whitehead J. H.

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1. Indeed, each open set in a locally compact space can be approximated from below by a sequence of compact sets, so that X − (G) ∈ F for all open G if and only if X − (K ) ∈ F for all K ∈ K. In a general Polish space other measurability deﬁnitions are possible. The following theorem of C. Himmelberg establishes the equivalence of several possible concepts. Its proof can be found in Himmelberg [257] and Castaing and Valadier [91]. 3 (Fundamental measurability theorem for multifunctions). Let E be a separable metric space.

Therefore, the inclusion functional I X (L), L ∈ I, and the capacity functional TX (L), L ∈ I, restricted to the family I of ﬁnite sets can be expressed from each other by solving systems of linear equations. By integrating the covariance function of X one obtains K (z) = Σ X (x, x + z)dx = P {{x, x + z} ⊂ X} dx = E mes(X ∩ (X − z)) , called the geometric covariogram of X. e. Σ X (x 1 , x 2 ) = Σ(x 1 − x 2 ). If X is a stationary isotropic random set, then Σ X (x 1 , x 2 ) depends only on r = x 1 − x 2 .

S. s. s. 2 Measurability and selections 39 Proof. Note that ϕ(ω, x) = (ω, ζx (ω)) is measurable with respect to the product σ -algebra F ⊗ B(E), whence {(ω, x) : ϕ(ω, x) ∈ B} ∈ F ⊗ B(E) for every B ∈ B(E ). The proof is ﬁnished by observing that Graph(Z ) = Graph(X) ∩ ϕ −1 (Graph(Y )) is a measurable subset of Ω × E. 2. 26 for the second one. 2. The minimum α and argmin inside X of a random function ζ . 27 (Measurability of inﬁmum). Let X be an almost surely non-empty random closed set in Polish space E and let ζx be an almost surely continuous stochastic process with values in R.