# 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  and Castaing and Valadier . 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.