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.