7. The Euler Characteristic 31 In general, we conclude that lim −→+0 χτ− (fτ− ) is the sum of αi such that Ai ∩ Hτ− = ∅ for all suﬃciently small 0. It follows then that χτ(fτ) − lim −→+0 χτ− (fτ− ) = i∈I αi, where I = i : min x∈Ai (x) = τ . In particular, lim −→+0 χτ− (fτ− ) = χτ(fτ) unless τ is the minimum value of the linear function (x) on some set Ai. Therefore, for a given function f ∈ K(Rd) there are only finitely many τ’s, where lim −→+0 χτ− (fτ− ) = χτ(fτ). Now we define χ(f) = τ∈R χτ(fτ) − lim −→+0 χτ− (fτ− ) . As we noted, the sum contains only finitely many non-zero summands, so it is well defined. If f, g ∈ K(Rd) are functions and α, β ∈ R are numbers, then for every τ ∈ R we have (αf + βg)τ = αfτ + βgτ. Since by the induction hypothesis χτ is a valuation and taking the limit is a linear operation, we conclude that χ(αf + βg) = αχ(f) + βχ(g), so χ is a valuation. Furthermore, if A ⊂ Rd is a compact convex set, then χτ([A]τ) − lim −→+0 χτ− ([A]τ− ) = 1 if minx∈A (x) = τ, 0 otherwise. Since A is a non-empty compact convex set, there is unique minimum value of the linear function (x) on A. Therefore, χ([A]) = 1. Now we are ready to extend χ onto C(Rd). Let B(ρ) = x ∈ Rd : x≤ ρ be the ball of radius ρ. For f ∈ C(Rd) we define χ(f) = lim ρ−→+∞ f · [B(ρ)]. Clearly, χ satisfies the required properties. Theorem 7.4 and its proof belongs to H. Hadwiger. If A ⊂ Rd is a set such that [A] ∈ C(Rd), we often write χ(A) instead of χ([A]) and call it the Euler characteristic of the set A. In the course of the proof of Theorem 7.4, we established the following useful fact, which will play the central role in our approach to the Euler-Poincar´ e Formula of Section VI.3. (7.5) Lemma. Let A ⊂ Rd be a set such that [A] ∈ K(Rd). For τ ∈ R let Hτ be the hyperplane consisting of the points x = (ξ1, . . . , ξd) with ξd = τ. Then [A ∩ Hτ] ∈ K(Rd) and χ(A) = τ∈R χ(A ∩ Hτ) − lim −→+0 χ(A ∩ Hτ− ) . Another useful result allows us to express the Euler characteristic of a union of sets in terms of the Euler characteristics of the intersections of the sets.

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