*(A, x)*→*(1{x* ∈*∂*^{+}*A*}*,***1{***x*∈*∂*^{++}*A*}*,***1{***x* ∈*∂*^{1}*A*}*, ν(A, x))*
*fromF** ^{d}*×R

^{d}*to*R×R×R×

*S*

^{d}^{−}

^{1}

*is 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* ⊂R

^{d}*,ε >0, for any measurable setD*⊂

*B*×

*S*

^{d}^{−}

^{1}

*, andj*∈ {0, . . . , d−1}

*the mapping*

*A*→*µ**j**(A*; {*(x, u)*∈*N (A)*:*(x, u)*∈*D, δ(A, x, u) > ε*}*)*

*fromF*^{d}*to*R*is measurable. Furthermore, the mapA* → |*µ** _{j}*|

*(A*; ·

*)fromF*

^{d}*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 where*µ*_{j}*(A*; ·*)*
has finite total variation. Let*C*0denote a countable and dense (with respect to the
maximum norm) set of continuous functions*f* :R* ^{d}*×

*S*

^{d}^{−}

^{1}→[0,1] with compact support. Then

|*µ** _{j}*|

*(A*;

*C)*=sup

*C*

*f (x, u)µ*_{j}*(A*;*d(x, u))*:*f* ∈*C*0

*,*

which yields the required measurability.

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

* Theorem 6.5. Let*V

*be aσ-finite measure onF*

^{d}*andB*⊂R

^{d}*a measurable set.*

*Then*

*F*^{d}

R* ^{d}*\A

**1**{0

*< d(A, z)*≤

*r, p(A, z)*∈

*B*}H

^{d}*(dz)*V

*(dA) <*∞ (6.1)

*for somer >0 if and only if*

*F*^{d}

R* ^{d}*×

*S*

^{d}^{−}

^{1}

**1{***x*∈*B*}*(δ(A, x, u)*∧*r)*^{d}^{−}* ^{j}*|

*µ*

*|*

_{j}*(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 constantsc*3*(l, d)*appearing in (2.21) do not depend on*A. Therefore,*
if (6.1) holds for some fixed*r* =*r*0 *>*0, then (6.2) holds with the same*r* =*r*0.
But then

*F*^{d}

R* ^{d}*×

*S*

^{d}^{−}

^{1}

**1{***ε*≤*δ(A, x, u), x*∈*B*}|*µ** _{j}*|

*(A*;

*d(x, u))*V

*(dA) <*∞ for

*j*=0, . . . , d−1, first for 0

*< ε < r*0and then for all

*ε >*0. Together with (6.2), for

*r*=

*r*0, this implies that (6.2) is true for all

*r >*0. Conversely, if (6.2) holds for just one

*r*

_{0}

*>*0 and for

*j*=0, . . . , d−1, then it holds for all

*r >*0 and

(6.1) follows from the local Steiner formula.

The inclusion

{*z*∈R* ^{d}* : 0

*< d(A, z)*≤

*r, p(A, z)*∈

*B*} ⊂

*A*

_{⊕}

*∩*

_{r}*B*

_{⊕}

_{r}implies that (6.1) is trivially satisfied if*B*is bounded andVis a finite measure.

**7 Contact distributions of random closed sets**

In this section, we consider a random closed set*Z*defined on the probability space
*(,A,*P*). Formally,* *Z* is a random element of the measurable space*F** ^{d}*. The
requirement

*Z*= ∅is no restriction of generality. If

*Z*is a random element of

*F*

*∪ {∅}, then we can apply the results of this section to the conditional prob-ability measure P*

^{d}*(*·|

*Z*= ∅

*). Our basic assumption onZ*

*is stationarity, i.e. the*distributional invariance of

*Z*under all translations.

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

of*Z*can be expressed as*p*=E[*H*^{d}*(Z*∩*B)] for any Borel setB*of volume 1. We
study the distribution of the distance*d(Z, z)*of*z* ∈ R* ^{d}* from

*Z. Ifp*=1, then P

*(d(Z, z)*=0)=1 for all

*z*∈R

*and hence*

^{d}*Z*=R

*is satisfiedP-almost surely.*

^{d}To exclude this trivial case we assume here that*p <* *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,*H*is independent of*z. 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 measurable*C* ⊂*S*^{d}^{−}^{1}.

It follows from Theorem 6.5 that E

**1**{*x* ∈*B*}*(δ(Z, x, u)*∧*r)*^{d}^{−}* ^{j}*|

*µ*

*j*|

*(Z*;

*d(x, u))*

*<*∞ (7.2)
for *j* = 0, . . . , d−1, for all compact sets*B* ⊂ R* ^{d}* and all

*r >*0. This is the expected value version of (2.2). Define

*β*_{j}*(*·*)*:=

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

Then*β*_{j}*(*R* ^{d}*×

*S*

^{d}^{−}

^{1}×{0}

*)*=0 so that

*β*

*can be interpreted as a random signed mea-sure onR*

_{j}*×*

^{d}*S*

^{d}^{−}

^{1}×

*(0,*∞]. Since

*Z*is assumed to be stationary, it can be easily seen from translation covariance and from the equation

*δ(Z*+

*y, x, u)*=

*δ(Z, x*−

*y, u),*

*y*∈R

*, that*

^{d}*β*

*is stationary, i.e. its distribution is invariant under shifts in the first variable. Therefore, if*

_{j}*B*⊂R

*is a Borel set with 0*

^{d}*<H*

^{d}*(B) <*∞, then

_{j}*(*·*)*:= 1
*H*^{d}*(B)*E

**1{***x*∈*B*}1{*(u, δ(Z, x, u))*∈ ·}*µ*_{j}*(Z*;*d(x, u))*

(7.3)

is a signed measure which does not depend on the choice of*B. In general we may*
have|*j**(S*^{d}^{−}^{1}×*(s,*∞])| → ∞as*s*→0, but equation (7.2) implies that

_{r}

0

*s*^{d}^{−}^{1}^{−}* ^{j}*|

*j*|

*(S*

^{d}^{−}

^{1}×

*(s,*∞])ds <∞ (7.4) for

*r*≥0 and

*j*=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 set

*Z*was assumed to be

*S*

*-valued. A first version of a result of this type has been established in [22].*

^{d}**Theorem 7.1. Let**Zbe a stationary random closed set. Then*(1*−*p)H (t, C)*=

*d*−1

*i*=0

*ω**d*−*i*

_{t}

0

*s*^{d}^{−}^{1}^{−}^{i}*i**(C*×*(s,*∞])ds
*for anyt*≥*0 and any measurable setC* ⊂*S*^{d}^{−}^{1}*.*

Absolute continuity of the contact distribution*H (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*(1*−*p)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 on*Z.*

Since_{i}*(C*×*(s,*∞])is a right continuous function of*s* ∈*(0,*∞*), it follows*
that*(1*−*p)H (*·*, C)*admits a right derivative on*(0,*∞*). Moreover, since the left*
limit of_{i}*(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*(1*−*p)H (*·*, C)*need not be differentiable at
the point 0. Such a differentiability property can be deduced under the additional
assumption

E[|*µ** _{i}*|

*(Z*;

*B*×

*S*

^{d}^{−}

^{1}

*)]<*∞

*,*

*i*=0, . . . , d−1, (7.5) for some Borel set

*B*with positive volume. Under this assumption (which certainly excludes fractal behaviour of

*Z), the curvature measuresµ*

_{i}*(Z*; ·

*)*can be considered as random signed measures onR

*×*

^{d}*S*

^{d}^{−}

^{1}. The associated total variation measures

|*µ** _{i}*|

*(Z*; ·

*)*are (locally finite) random measures.

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

*t*→lim0+*t*^{−}^{1}*(1*−*p)H (t, C)*=2λ*d*−1*(C)*
*for any measurable setC*⊂*S*^{d}^{−}^{1}*, where*

*λ*_{d}_{−}_{1}*(C)*:=E[µ*d*−1*(Z*;[0,1]* ^{d}*×

*C)]<*∞

*.*(7.6)

*Proof. Assumption (7.5) ensures that the*

*are finite signed measures. Therefore the assertion is an immediate consequence of Theorem 7.1.*

_{j}Under (7.5) we have in particular that

*d*−1:=E[µ*d*−1*(Z*; ·*)]* (7.7)
is a locally finite measure onR* ^{d}* ×

*S*

^{d}^{−}

^{1}. According to Proposition 4.1 we may interpret

*d*−1

*(*· ×

*S*

^{d}^{−}

^{1}

*)as the surface intensity measure ofZ. From stationarity*we obtain that

*d*−1=*H** ^{d}*⊗

*λ*

*d*−1

*.*(7.8)

Again by Proposition 4.1 we can interpret the number*λ*_{d}_{−}_{1}*(S*^{d}^{−}^{1}*)as the surface*
*intensity ofZ*and (assuming*λ*_{d}_{−}_{1}*(S*^{d}^{−}^{1}*) >*0) the probability measure

*R*:=*λ*_{d}_{−}_{1}*/λ*_{d}_{−}_{1}*(S*^{d}^{−}^{1}*)*
*as the rose of directions ofZ.*

In some applications one might have*λ*_{d}_{−}_{1}*(S*^{d}^{−}^{1}*)*= 0. An example are fibre
processes inR^{3}(see [31]). If the surface intensity is 0, the first (right) derivative of
*(1*−*p)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. Let**Zbe a non-empty stationary random closed set satisfying (7.5)*andλ**d*−1*(S*^{d}^{−}^{1}*)*=*0. Then, for any measurable setC* ⊂ *S*^{d}^{−}^{1}*,(1*−*p)H (*·*, C)*
*has a second derivative at the point 0 which is given by*

2πE[µ*d*−2*(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

*(Z** _{n}*+

*ξ*

_{n}*),*

where the *ξ**n**, n* ∈ N, build a stationary Poisson process inR* ^{d}* with positive
and finite intensity

*γ*

*and where the grainsZ*1

*, Z*2

*, . . .*form a sequence of inde-pendent, identically distributed random elements of

*C*

*(the space of non-empty compact subsets ofR*

^{d}*) which is independent of (see [23] and [31] for more details). Denoting the common distribution of the*

^{d}*Z*

*byQwe make the standard assumption*

_{n}

*H*^{d}*(A*⊕*B)*Q*(dA) <*∞ (8.1)

for all compact sets*B*⊂R* ^{d}*, where

*A*⊕

*B*:= {

*x*+

*y*:

*x*∈

*A, y*∈

*B*}. Since any bounded set can be covered by finitely many balls of a fixed radius, condition (8.1) is equivalent to

*H*^{d}*(A*_{⊕}_{r}*)*Q*(dA) <*∞ (8.2)
for just one*r >*0. Assumption (8.1) guarantees that each compact set is intersected
by only a finite number of the (shifted) grains*Z** _{n}*+

*ξ*

*,*

_{n}*n*∈N.

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

P*(Z*∩*B*= ∅*)*=1−exp
strictly less than 1, if*B*is bounded. In particular, we obtain for the volume fraction
that

Taking*B*=*B** ^{d}*in (8.3) and using the preceding formula for

*p, we obtain*1−

*H (t )*=exp

By Theorem 6.5, assumption (8.1) implies that

*(δ(A, x, u)*∧*r)*^{d}^{−}* ^{j}*|

*µ*

*|*

_{j}*(A*;

*d(x, u))*Q

*(dA) <*∞ (8.5) for

*r >*0 and

*j*=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
functions*H (*·*, C).*

* Theorem 8.1. LetZbe the stationary Boolean model defined above and letC* ⊂

*S*

^{d}^{−}

^{1}

*be measurable. ThenH (*·

*, C)is absolutely continuous with density*

*t* →*(1*−*H (t ))γ*

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

*(1*−*H (d(A, z)))1*{0*< d(A, z)*≤*t, u(A, z)*∈*C*}

×H^{d}*(dz)*Q*(dA)* (8.6)

for all*t* ≥0 and all measurable*C* ⊂*S*^{d}^{−}^{1}. 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 *A*_{⊕}* _{s}* has volume
0, for all

*A*∈

*C*

*and*

^{d}*s >*0. This can be seen as follows. Let

*x*∈

*∂A*

_{⊕}

*be fixed for the moment. Then there is some*

_{s}*a*∈ cl

*A*such that|

*x*−

*a*| =

*s*and

*∂A*_{⊕s} ∩int*B*^{d}*(a, s)*= ∅. Therefore, we find that
lim sup

*r*→0+

*H*^{d}*(∂A*_{⊕}* _{s}*∩

*B*

^{d}*(x, r))*

*H*

^{d}*(B*

^{d}*(x, r))*

*<*1,

for any*x*∈*∂A*_{⊕}* _{s}*. The assertion now follows from Theorem 2.9.11 in [5].

Since*z* → 1−*H (d(A, z))*is a bounded function and*A*is 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 that*Qis the distribution of a random multiple*ξ A*0of some
fixed compact set *A*0, where*ξ* is a positive random variable withE[ξ* ^{d}*]

*<*∞. Using the scaling properties in Proposition 4.9 we obtain from Theorem 8.1 that

*H (t, C)*has the density

*(1*−*H (t ))γ*

*d*−1

*i*=0

*ω*_{d}_{−}_{i}*t*^{d}^{−}^{1}^{−}^{i}

×E

*ξ*^{i}

**1**{*t < ξ δ(A*0*, x, u)*}**1**{*u*∈*C*}*µ**i**(A*0;*d(x, u))*

*.*

If*A*0is a fractal, we might have that the above density tends to∞as*t* →0. Indeed,
it follows directly from (8.4) that

*H (t )*=1−exp

−*γ*E[ξ^{d}*H*^{d}*(A*_{0}+*ξ*^{−}^{1}*t B*^{d}*)]*

*.*

Assume now that*A*_{0}is a fractal with box-counting dimension*a*∈*(d*−1, d). For
instance*A*_{0}could be the Sierpi´nski gasket introduced and discussed in Example 4.8.

Further, we assume that*ξ* ≥*t*_{0}*>*0 holdsP-a.s. Then we choose*ε*∈*(0, a*−*d*+1).

By definition, if*t >*0 is sufficiently small, we get that
*d*−*a*+*ε >* log*H*^{d}*(A*0+*ξ*^{−}^{1}*t B*^{d}*)*

log(ξ^{−}^{1}*t )* *,*
hence

*ξ*^{d}*H*^{d}*(A*_{0}+*ξ*^{−}^{1}*t B*^{d}*) > t*^{d}^{−}^{a}^{+}^{ε}*t*_{0}^{a}^{−}^{ε}*.*

This shows that

*H (t )*≥*(γ /2)t*_{0}^{a}^{−}^{ε}*t*^{d}^{−}^{a}^{+}^{ε}

if*t >*0 is sufficiently small. In particular,*t*^{−}^{1}*H (t )*→ ∞as*t*→0.

Finally, we turn to the relationships between the measures* _{j}* introduced in
Section 7 for a general stationary closed set and corresponding mean values with
respect toQ.

**Theorem 8.3. Let**Zbe a stationary Boolean model as above. Then_{j}*(C*×*(s,*∞])=*(1*−*p)(1*−*H (s))γ*

**1{***s < δ(A, x, u)*}

×**1{***u*∈*C*}*µ*_{j}*(A*;*d(x, u))*Q*(dA),* (8.7)
*for all measurable setsC*⊂*S*^{d}^{−}^{1}*,s >0, andj* ∈ {0, . . . , d−1}.

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

* Lemma 8.4. LetA*⊂R

^{d}*be the union set of the locally finite family of compact sets*

*A*

*⊂R*

_{i}

^{d}*,i*∈N. Let

*j*∈ {0, . . . , d−1},

*s >0, and letB*⊂R

^{d}*be measurable and*

*bounded. Then, for all measurable and bounded functionsf*:R

*×*

^{d}*S*

^{d}^{−}

^{1}→R,

*f (x, u)1*{*s < δ(A, x, u), x*∈*B*}*µ**j**(A*;*d(x, u))*

=

*v*∈*S(*N*)*

**1{***B*^{d}*(x*+*su, s)*∩*A** ^{(v)}*= ∅}

*i*∈*v*

**1{***s < δ(A*_{i}*, x, u), x*∈*B*}

×*f (x, u)µ**j**(A**v*;*d(x, u)),*
*where*

*A** _{v}*:=

*i*∈*v*

*A*_{i}*and* *A** ^{(v)}*:=

*i /*∈*v*

*A*_{i}*.*

*An analogous relationship is satisfied for the total variation measures*|*µ** _{j}*|

*(A*; ·

*)*

*and*|

*µ*

*|*

_{j}*(A*

*; ·*

_{v}*),v*∈

*S(*N

*).*

*Proof. It is an easy consequence of the definition thatN (A)*is the disjoint union
of the sets*D**v*,*v*∈*S(*N*), where*

*D** _{v}*:=

*((*R

*\*

^{d}*A*

^{(v)}*)*×

*S*

^{d−}^{1}

*)*∩

*i*∈*v*

*N (A*_{i}*).*

Moreover, if*(x, u)* ∈ *D** _{v}*and

*s >*0, then

*δ(A, x, u) > s*if and only if

*B*

^{d}*(x*+

*su, s)*∩

*A*

*= ∅and*

^{(v)}*δ(A*

_{i}*, x, u) > s*for

*i*∈

*v. Since clearly*

*i*∈*v**N (A*_{i}*)* ⊂
*N (A**v**), we have (in obvious notation)H**d*−1−*j**(A, x, u)*=*H**d*−1−*j**(A**v**, x, u), for*
*H*^{d}^{−}^{1}-a.e.*(x, u)*∈*D**v*. Hence the result follows from (2.24).

*Proof of Theorem 8.3. LetB*⊂R* ^{d}*denote 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
where*dx*_{1}*, . . . , dx** _{n}*denote integration with respect to Lebesgue measure. Using
this result for

*C*=

*S*

^{d}^{−}

^{1}and comparing Theorem 8.1 with Theorem 7.1, we obtain that

for all*(x, u)*∈*N (B*1*)*∩*. . .*∩*N (B*_{n}*). Therefore we can apply (2.21) to obtain from*
(8.9) that

0=^{∞}

*n*=2

*γ*^{n}*n!*

_{r}

0

*s*^{d}^{−}^{1}^{−}^{k}

· · · ·

*B*×*S*^{d}^{−}^{1}

×**1{***s < δ*^{}*(A*_{1}+*x*_{1}*, . . . , A** _{n}*+

*x*

_{n}*, x, u)*}

× |*µ** _{k}*|

*((A*

_{1}+

*x*

_{1}

*)*∩

*. . .*∩

*(A*

*+*

_{n}*x*

_{n}*)*;

*d(x, u))*

×*dx*1*. . . dx** _{n}*Q

*(dA*1

*) . . .*Q

*(dA*

_{n}*)ds*for all

*k*∈ {0, . . . , d−1}and all

*r >*0. Hence

0=

· · · ·

*B×S*^{d−1}

×**1**{*s < δ*^{}*(A*1+*x*1*, . . . , A**n*+*x**n**, x, u)*}

× |*µ** _{k}*|

*((A*1+

*x*1

*)*∩

*. . .*∩

*(A*

*+*

_{n}*x*

_{n}*)*;

*d(x, u))*

×*dx*_{1}*. . . dx** _{n}*Q

*(dA*

_{1}

*) . . .*Q

*(dA*

_{n}*)*

for all*n*≥2,*k*∈ {0, . . . , d−1}and*s >*0. Inserting this relation into (8.8), we

obtain the desired result.

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

|*µ** _{j}*|

*(A*;R

*×*

^{d}*S*

^{d}^{−}

^{1}

*)*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[µ*i**(Z*; ·*)],* *i*=0, . . . , d−1,
are signed Radon measures onR* ^{d}*×

*S*

^{d}^{−}

^{1}. Since

*Z*is stationary,

*i* =*H** ^{d}*⊗

*λ*

*i*

*,*

*i*=0, . . . , d−1, (8.11) where

*λ*

_{i}*(C)*=

_{i}*([0,*1]

*×*

^{d}*C)*for any Borel set

*C*⊂

*S*

^{d}^{−}

^{1}. Note that

*λ*

_{d}_{−}

_{1}has already been introduced by (7.8).

Together with

*H*^{d}*(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. Let**Z*be the stationary Boolean model defined above and assume*
*that (8.10) holds. Then (7.5) is satisfied and*

*λ*_{j}*(C)*=*(1*−*p)γ*

*µ*_{j}*(A*;R* ^{d}*×

*C)*Q

*(dA)*(8.12)

*forj*=0, . . . , d−

*1 and all measurable setsC*⊂

*S*

^{d}^{−}

^{1}

*.*

*Proof. Fix a Borel setC* ⊂*S*^{d}^{−}^{1}and*j* ∈ {0, . . . , d−1}, and let*B* ⊂R* ^{d}* be a
bounded and measurable set of volume 1. It follows as in the proof of Theorem 8.3
that

E

**1{***x* ∈*B*}1{*s < δ(Z, x, u)*}|*µ** _{j}*|

*(Z*;

*d(x, u))*

=*(1*−*p)(1*−*H (s))γ*

**1{***s < δ(A, x, u)*}|*µ** _{j}*|

*(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

*λ*_{j}*(C)*=E[µ*j**(Z*;*B*×*C)]*=_{j}*(C*×*(0,*∞]). (8.14)
Letting*s* →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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