(A, x)(1{x+A},1{x++A},1{x1A}, ν(A, x)) fromFd×RdtoR×R×R×Sd1is measurable and covariant.

The proofs of the preceding three lemmas follow from the results and arguments provided at the end of Section 3 in [17].

Lemma 6.4. For any compact setB ⊂Rd,ε >0, for any measurable setDB×Sd1, andj ∈ {0, . . . , d−1}the mapping

Aµj(A; {(x, u)N (A):(x, u)D, δ(A, x, u) > ε})

fromFdtoRis measurable. Furthermore, the mapA → |µj|(A; ·)fromFd to [0,∞] is measurable.

Proof. The first assertion is implied by Lemmas 6.1 and 6.2, Fubini’s theorem and by relation (2.26).

For the second assertion, it is sufficient to consider the case whereµj(A; ·) has finite total variation. LetC0denote a countable and dense (with respect to the maximum norm) set of continuous functionsf :Rd×Sd1→[0,1] with compact support. Then

|µj|(A;C)=sup

C

f (x, u)µj(A;d(x, u)):fC0

,

which yields the required measurability.

The following improvement of the integrability property (2.2) will be useful in the next section.

Theorem 6.5. LetVbe aσ-finite measure onFdandB ⊂Rda measurable set.

Then

Fd

Rd\A1{0< d(A, z)r, p(A, z)B}Hd(dz)V(dA) <∞ (6.1) for somer >0 if and only if

Fd

Rd×Sd1

1{xB}(δ(A, x, u)r)dj|µj|(A;d(x, u))V(dA) <∞ (6.2) for somer >0 and allj = 0, . . . , d−1. In this case, both (6.1) and (6.2) are satisfied by anyr >0.

Proof. The constantsc3(l, d)appearing in (2.21) do not depend onA. Therefore, if (6.1) holds for some fixedr =r0 >0, then (6.2) holds with the samer =r0. But then

Fd

Rd×Sd1

1{εδ(A, x, u), xB}|µj|(A;d(x, u))V(dA) <∞ forj =0, . . . , d−1, first for 0 < ε < r0and then for allε >0. Together with (6.2), forr =r0, this implies that (6.2) is true for allr >0. Conversely, if (6.2) holds for just oner0>0 and forj =0, . . . , d−1, then it holds for allr >0 and

(6.1) follows from the local Steiner formula.

The inclusion

{z∈Rd : 0< d(A, z)r, p(A, z)B} ⊂ArBr

implies that (6.1) is trivially satisfied ifBis bounded andVis a finite measure.

7 Contact distributions of random closed sets

In this section, we consider a random closed setZdefined on the probability space (,A,P). Formally, Z is a random element of the measurable spaceFd. The requirement Z = ∅is no restriction of generality. IfZ is a random element of Fd∪ {∅}, then we can apply the results of this section to the conditional prob-ability measure P(·|Z = ∅). Our basic assumption onZ is stationarity, i.e. the distributional invariance ofZunder all translations.

Due to stationarity, the volume fraction p:=P(0Z)

ofZcan be expressed asp=E[Hd(ZB)] for any Borel setBof volume 1. We study the distribution of the distanced(Z, z)ofz ∈ Rd fromZ. Ifp =1, then P(d(Z, z)=0)=1 for allz∈Rdand henceZ =Rdis satisfiedP-almost surely.

To exclude this trivial case we assume here thatp < 1. The spherical contact distribution function ofZ(see e.g. [31]) is defined by

H (t ):=P(d(Z, z)t|z /Z), t ≥0.

Again by stationarity,His independent ofz. More generally, we define (see [22], [17], [19])

H (t, C):=P(d(Z, z)t, u(Z, z)C|z /Z), (7.1) for any measurableCSd1.

It follows from Theorem 6.5 that E

1{xB}(δ(Z, x, u)r)dj|µj|(Z;d(x, u))

<∞ (7.2) for j = 0, . . . , d−1, for all compact setsB ⊂ Rd and allr > 0. This is the expected value version of (2.2). Define

βj(·):=

1{(x, u, δ(Z, x, u))∈ ·}µj(Z;d(x, u)), j =0, . . . , d−1.

Thenβj(Rd×Sd1×{0})=0 so thatβjcan be interpreted as a random signed mea-sure onRd×Sd1×(0,∞]. SinceZis assumed to be stationary, it can be easily seen from translation covariance and from the equationδ(Z+y, x, u)=δ(Z, xy, u), y ∈Rd, thatβj is stationary, i.e. its distribution is invariant under shifts in the first variable. Therefore, ifB⊂Rdis a Borel set with 0<Hd(B) <∞, then

j(·):= 1 Hd(B)E

1{xB}1{(u, δ(Z, x, u))∈ ·}µj(Z;d(x, u))

(7.3)

is a signed measure which does not depend on the choice ofB. In general we may have|j(Sd1×(s,∞])| → ∞ass→0, but equation (7.2) implies that

r

0

sd1j|j|(Sd1×(s,∞])ds <∞ (7.4) forr≥0 andj =0, . . . , d−1. The following result can now be proved similarly to Theorem 5.1 and Corollary 5.2 in [17]. In that paper the random closed setZwas assumed to beSd-valued. A first version of a result of this type has been established in [22].

Theorem 7.1. LetZbe a stationary random closed set. Then (1p)H (t, C)=

d1

i=0

ωdi

t

0

sd1ii(C×(s,∞])ds for anyt0 and any measurable setCSd1.

Absolute continuity of the contact distributionH (t ) has been proved (inde-pendently of [22]) in [2] (see also [13]) using Federer’s coarea theorem. The new and very pleasing fact here is that the density of(1p)H (·, C) is of the same explicit form as in [22] and [17] where the case of a random set taking values in the extended convex ring is studied. Note in particular that we do not need to impose any integrability condition onZ.

Sincei(C×(s,∞])is a right continuous function ofs(0,), it follows that(1p)H (·, C)admits a right derivative on(0,). Moreover, since the left limit ofi(C×(·,∞])exists, the contact distribution also has a left derivative.

It is even differentiable with the possible exception of at most countably many points. Example 8.2 below shows that(1p)H (·, C)need not be differentiable at the point 0. Such a differentiability property can be deduced under the additional assumption

E[|µi|(Z;B×Sd1)]<, i=0, . . . , d−1, (7.5) for some Borel setBwith positive volume. Under this assumption (which certainly excludes fractal behaviour ofZ), the curvature measuresµi(Z; ·)can be considered as random signed measures onRd×Sd1. The associated total variation measures

|µi|(Z; ·)are (locally finite) random measures.

Corollary 7.2. LetZbe a stationary random closed set satisfying (7.5) for some Borel setBwith positive volume. Then

tlim0+t1(1p)H (t, C)=2λd1(C) for any measurable setCSd1, where

λd1(C):=E[µd1(Z;[0,1]d×C)]<. (7.6) Proof. Assumption (7.5) ensures that thej are finite signed measures. Therefore the assertion is an immediate consequence of Theorem 7.1.

Under (7.5) we have in particular that

d1:=E[µd1(Z; ·)] (7.7) is a locally finite measure onRd ×Sd1. According to Proposition 4.1 we may interpretd1(· ×Sd1)as the surface intensity measure ofZ. From stationarity we obtain that

d1=Hdλd1. (7.8)

Again by Proposition 4.1 we can interpret the numberλd1(Sd1)as the surface intensity ofZand (assumingλd1(Sd1) >0) the probability measure

R:=λd1d1(Sd1) as the rose of directions ofZ.

In some applications one might haveλd1(Sd1)= 0. An example are fibre processes inR3(see [31]). If the surface intensity is 0, the first (right) derivative of (1p)H (·, C)vanishes at the point 0. Since the first derivative is itself differen-tiable with the exception of only countably many points, we then can consider the second (right) derivative at 0. This yields the following result.

Corollary 7.3. LetZbe a non-empty stationary random closed set satisfying (7.5) andλd1(Sd1)=0. Then, for any measurable setCSd1,(1p)H (·, C) has a second derivative at the point 0 which is given by

2πE[µd2(Z;[0,1]d×C)].

8 Contact distributions of Boolean models

We finally discuss the important special case of a Boolean model with compact particles. Hence we assume now that

Z=

n∈N

(Zn+ξn),

where the ξn, n ∈ N, build a stationary Poisson process inRd with positive and finite intensityγ and where the grainsZ1, Z2, . . . form a sequence of inde-pendent, identically distributed random elements of Cd (the space of non-empty compact subsets ofRd) which is independent of (see [23] and [31] for more details). Denoting the common distribution of theZnbyQwe make the standard assumption

Hd(AB)Q(dA) <∞ (8.1)

for all compact setsB⊂Rd, whereAB := {x+y:xA, yB}. Since any bounded set can be covered by finitely many balls of a fixed radius, condition (8.1) is equivalent to

Hd(Ar)Q(dA) <∞ (8.2) for just oner >0. Assumption (8.1) guarantees that each compact set is intersected by only a finite number of the (shifted) grainsZn+ξn,n∈N.

A good starting point for the analysis of the spherical contact distribution func-tion is the formula for the capacity funcfunc-tional ofZstating that

P(ZB= ∅)=1−exp strictly less than 1, ifBis bounded. In particular, we obtain for the volume fraction that

TakingB=Bdin (8.3) and using the preceding formula forp, we obtain 1−H (t )=exp

By Theorem 6.5, assumption (8.1) implies that

(δ(A, x, u)r)dj|µj|(A;d(x, u))Q(dA) <∞ (8.5) forr >0 andj =0, . . . , d−1. Therefore we can apply the local Steiner formula (2.3) to deduce from (8.4) that

H (t )=1−exp

Using different methods we can generalize this result to contact distribution functionsH (·, C).

Theorem 8.1. LetZbe the stationary Boolean model defined above and letCSd1be measurable. ThenH (·, C)is absolutely continuous with density

t(1H (t ))γ

Proof. It follows exactly as in the proof of Theorem 3.1 in [18] that H (t, C)=γ

(1H (d(A, z)))1{0< d(A, z)t, u(A, z)C}

×Hd(dz)Q(dA) (8.6)

for allt ≥0 and all measurableCSd1. The only additional argument, which is required, concerns Lemma 3.1 of [18]. In order to get the corresponding result in the present context, we need to show that the boundary of As has volume 0, for all ACd ands > 0. This can be seen as follows. Let x∂As be fixed for the moment. Then there is some a ∈ clAsuch that|xa| = s and

∂A⊕s ∩intBd(a, s)= ∅. Therefore, we find that lim sup

r0+

Hd(∂AsBd(x, r)) Hd(Bd(x, r)) <1,

for anyx∂As. The assertion now follows from Theorem 2.9.11 in [5].

Sincez → 1−H (d(A, z))is a bounded function andAis compact, we can exploit the local Steiner formula to express the inner integral of (8.6) in terms of the support measures. By (8.5) we can then use Fubini’s theorem to conclude the

desired result.

Example 8.2. Assume thatQis the distribution of a random multipleξ A0of some fixed compact set A0, whereξ is a positive random variable withE[ξd] < ∞. Using the scaling properties in Proposition 4.9 we obtain from Theorem 8.1 that H (t, C)has the density

(1H (t ))γ

d1

i=0

ωditd1i

×E

ξi

1{t < ξ δ(A0, x, u)}1{uC}µi(A0;d(x, u))

.

IfA0is a fractal, we might have that the above density tends to∞ast →0. Indeed, it follows directly from (8.4) that

H (t )=1−exp

γE[ξdHd(A0+ξ1t Bd)]

.

Assume now thatA0is a fractal with box-counting dimensiona(d−1, d). For instanceA0could be the Sierpi´nski gasket introduced and discussed in Example 4.8.

Further, we assume thatξt0>0 holdsP-a.s. Then we chooseε(0, ad+1).

By definition, ift >0 is sufficiently small, we get that da+ε > logHd(A0+ξ1t Bd)

log(ξ1t ) , hence

ξdHd(A0+ξ1t Bd) > tda+εt0aε.

This shows that

H (t )(γ /2)t0aεtda+ε

ift >0 is sufficiently small. In particular,t1H (t )→ ∞ast→0.

Finally, we turn to the relationships between the measuresj introduced in Section 7 for a general stationary closed set and corresponding mean values with respect toQ.

Theorem 8.3. LetZbe a stationary Boolean model as above. Then j(C×(s,∞])=(1p)(1H (s))γ

1{s < δ(A, x, u)}

×1{uC}µj(A;d(x, u))Q(dA), (8.7) for all measurable setsCSd1,s >0, andj ∈ {0, . . . , d−1}.

The proof of this theorem requires the following lemma. Recall thatS(N) de-notes the system of all non-empty finite subsets ofN.

Lemma 8.4. LetA⊂Rdbe the union set of the locally finite family of compact sets Ai ⊂Rd,i∈N. Letj ∈ {0, . . . , d−1},s >0, and letB⊂Rdbe measurable and bounded. Then, for all measurable and bounded functionsf :Rd×Sd1→R,

f (x, u)1{s < δ(A, x, u), xB}µj(A;d(x, u))

=

vS(N)

1{Bd(x+su, s)A(v)= ∅}

iv

1{s < δ(Ai, x, u), xB}

×f (x, u)µj(Av;d(x, u)), where

Av:=

iv

Ai and A(v):=

i /v

Ai.

An analogous relationship is satisfied for the total variation measures|µj|(A; ·) and|µj|(Av; ·),vS(N).

Proof. It is an easy consequence of the definition thatN (A)is the disjoint union of the setsDv,vS(N), where

Dv:=((Rd\A(v))×Sd−1)

iv

N (Ai).

Moreover, if(x, u)Dvands > 0, thenδ(A, x, u) > sif and only ifBd(x+ su, s)A(v) = ∅andδ(Ai, x, u) > s foriv. Since clearly

ivN (Ai)N (Av), we have (in obvious notation)Hd1j(A, x, u)=Hd1j(Av, x, u), for Hd1-a.e.(x, u)Dv. Hence the result follows from (2.24).

Proof of Theorem 8.3. LetB⊂Rddenote a bounded Borel set of volume 1. expectation in deducing the previous equation, one first derives the corresponding equation for the total variation measure|j|, which is finite for the sets considered.

By a fundamental property of the Poisson process (see e.g. [20]) we conclude that wheredx1, . . . , dxndenote integration with respect to Lebesgue measure. Using this result forC =Sd1and comparing Theorem 8.1 with Theorem 7.1, we obtain that

for all(x, u)N (B1). . .N (Bn). Therefore we can apply (2.21) to obtain from (8.9) that

0=

n=2

γn n!

r

0

sd1k

· · · ·

B×Sd1

×1{s < δ(A1+x1, . . . , An+xn, x, u)}

× |µk|((A1+x1). . .(An+xn);d(x, u))

×dx1. . . dxnQ(dA1) . . .Q(dAn)ds for allk∈ {0, . . . , d−1}and allr >0. Hence

0=

· · · ·

B×Sd−1

×1{s < δ(A1+x1, . . . , An+xn, x, u)}

× |µk|((A1+x1). . .(An+xn);d(x, u))

×dx1. . . dxnQ(dA1) . . .Q(dAn)

for alln≥2,k∈ {0, . . . , d−1}ands >0. Inserting this relation into (8.8), we

obtain the desired result.

It is tempting to take the limits → 0 in (8.7). This requires the integrability condition

|µj|(A;Rd×Sd1)Q(dA) <, j =0, . . . , d−1. (8.10) As we will see, assumption (8.10) implies that (7.5) is satisfied (the converse is also true). Hence, under this condition

i :=E[µi(Z; ·)], i=0, . . . , d−1, are signed Radon measures onRd×Sd1. SinceZis stationary,

i =Hdλi, i=0, . . . , d−1, (8.11) whereλi(C)=i([0,1]d×C)for any Borel setCSd1. Note thatλd1has already been introduced by (7.8).

Together with

Hd(A)Q(dA) <∞, condition (8.10) is a stronger assump-tion than (8.1). Fractal grains are excluded this way. Our final theorem generalizes results in [23] and [17].

Theorem 8.5. LetZ be the stationary Boolean model defined above and assume that (8.10) holds. Then (7.5) is satisfied and

λj(C)=(1p)γ

µj(A;Rd×C)Q(dA) (8.12) forj =0, . . . , d−1 and all measurable setsCSd1.

Proof. Fix a Borel setCSd1andj ∈ {0, . . . , d−1}, and letB ⊂Rd be a bounded and measurable set of volume 1. It follows as in the proof of Theorem 8.3 that

E

1{xB}1{s < δ(Z, x, u)}|µj|(Z;d(x, u))

=(1p)(1H (s))γ

1{s < δ(A, x, u)}|µj|(A;d(x, u))Q(dA), (8.13) for all s > 0. Letting s → 0 we obtain (7.5) by dominated convergence. The assumption of stationarity implies that

λj(C)=E[µj(Z;B×C)]=j(C×(0,∞]). (8.14) Lettings →0 in (8.7), the assertion (8.12) now follows again by dominated

con-vergence.

Acknowledgements. Parts of this paper were written while the second author had been visiting the Australian National University in Canberra, supported by the Volkswagen Foundation. Thanks are also due to Dr. Jan Rataj for helpful comments on Lemma 2.2.

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