REPRESENTATION OF POTENTIALS

WOLFHARD HANSEN and IVAN NETUKA

Given a balayage space (X,W) and a Borel measurable (Green) function G : X ×X → [0,∞] which is locally bounded o the diagonal such that each G(·, y), y ∈ Y, is a potential with superharmonic support {y}, and each functionG(x,·),x∈X, is lower semicontinuous onXand continuous onX\ {x}

(even continuous onXifxis nely isolated), the following stability result for po-
tentialsG^{ν}:=R

G(·, y)dν(y)is obtained: Ifpis a potential on X such that, for
some sequence (µn) of (positive Radon) measures on X, the sequence (G^{µ}^{n})
is bounded by a potential and converges pointwise to p outside a polar set
(a semipolar set, if (X,W) is a harmonic space), then there exists a (unique)
measureµonXsuch thatp=G^{µ}(andµis the weak limit of the sequence(µn)).

An application characterizes the situation, where every potential on X has the
formG^{µ}.

AMS 2010 Subject Classication: 31D05.

Key words: potentials, convergence of potentials, Green function, representation of potentials, balayage spaces, harmonic spaces.

1. INTRODUCTION

Representation of potentials, as one of the important parts of classical as well as abstract potential theory, has attracted the attention of mathemati- cians for a long time. In classical potential theory representation of potentials goes back to [22]. Stability of representation with respect to increasing lim- its of Newtonian and Riesz potentials is established in ([16], Theorem 1.26, Theorem 3.9). Convergence properties of potentials of measures with respect to general kernels on locally compact spaces are investigated in [24]. In ab- stract potential theory, such results frequently rely on Choquet's theorem (see, for instance, [4, 6, 10, 18, 19, 25]). For representation theorems for excessive functions, see [2, 11, 15, 21].

The question of representability is closely related to the existence of a Green function, the axiom of proportionality, duality theory, and convergence axioms [1, 5, 7, 8, 10, 12, 14, 20, 23].

REV. ROUMAINE MATH. PURES APPL. 59 (2014), 1, 93104

The basic frame for the considerations in this paper is a balayage space (X,W), whereX is a locally compact space with countable base and W is the set of all positive hyperharmonic functions on X (see [3]). While the subclass ofP-harmonic spaces is designed for a unied discussion of solutions to various types of linear elliptic and parabolic partial dierential equations of second order, the notion of a balayage space covers, in addition, Riesz potentials, Markov chains on discrete spaces, and integro-dierential equations. In contrast to harmonic spaces, where harmonic measures for open sets are concentrated on the boundary, harmonic measures for open sets in balayage spaces may live on the entire complement. Nevertheless, the potential theory on balayage spaces is as rich as that on harmonic spaces.

Moreover, the study of a balayage space (X,W) (with 1 ∈ W, which up to normalization is no restriction) amounts to the study of the set of excessive functions for a sub-Markov resolvent V = (Vλ)λ>0 (a sub-Markov semigroup P= (Pt)t>0, a Hunt process X), where all excessive functions are lower semi- continuous and there are suciently many continuous ones.

More precisely,(X,W) is a balayage space, ifW is the set of all excessive
functions for a right continuous sub-Markov resolvent V= (V_{λ})_{λ>0} on X such
that its potential kernel V0 := sup_{λ>0}Vλ is proper, every function in W is the
supremum of its continuous minorants in W, and there exist strictly positive,
nite, and continuousu, v∈ W such thatu/v vanishes at innity.

Conversely, given any bounded continuous strict potentialpon a balayage
space(X,W)with1∈ W(they exist in profusion), there exists a (unique) sub-
Markov resolvent V= (V_{λ})_{λ>0} with p=V_{0}1, a unique sub-Markov semigroup
P= (Pt)t>0 such thatp =R∞

0 Pt1dt, and an associated Hunt process X such thatW is the set of all excessive functions with respect to V, P, and X.

In the following let (X,W) be a balayage space. We recall that the (W-)ne topology is the coarsest topology, for which all functions in W are continuous; it is ner (but maybe not strictly ner) than the initial topology on X.

Let us assume that there is a Borel measurable function G:X×X→[0,∞]

having the following three properties (for the denition of potentials and su- perharmonic support see Section 1):

(i) Gis locally bounded o the diagonal.

(ii) Each functionG(·, y) is a potential with superharmonic support {y}. (iii) Each function G(x,·) is lower semicontinuous on X and continuous on

X\ {x}.

Ifx∈X is nely isolated (and not isolated), then G(x,·)is even contin- uous on X.

Let M(X) be the set of all positive Radon measures on X. For every

µ∈ M(X), we dene

(1.1) G^{µ}:=

Z

G(·, y)dµ(y).

The main purpose of this paper is to show that, essentially, the set of all
representable potentials p, that is, such that p = G^{µ} for some µ ∈ M(X), is
stable under pointwise limits. More precisely, we have the following stability
result.

Theorem 1.1. Letpbe a potential onXsuch that, for some sequence(µn)
in M(X), the functions G^{µ}^{n}, n ∈ N, are bounded by some potential q on X
and converge pointwise to p outside a polar set (a semipolar set, if (X,W) is
a harmonic space).

Then there exists a unique µ ∈ M(X) such that p = G^{µ}. In fact, µ is
the weak limit of the sequence (µn) (that is, limn→∞µn(ϕ) = µ(ϕ) for every
ϕ∈ K(X)).

Remark 1.2. Let (µn) be a sequence in M(X). Of course, the assump-
tions of the theorem are satised ifp is a potential andG^{µ}^{n} ↑p. Moreover, the
assumptions are satised, if (X,W) is a harmonic space, (G^{µ}^{n}) is a decreas-
ing sequence of potentials, and p is the lower semicontinuous regularization
of infG^{µ}^{n}.

An application of Theorem 1.1 (see Theorem 4.1) yields various character- izations of the possibility to represent all potentials (which, partly under more restrictive assumptions, partly with more complicated proofs, have already been obtained before, see [13, 17]).

2. PRELIMINARIES

Let us rst introduce some standard notations. Let C(X) denote the set
of all nite and continuous functions onX. LetK(X)be the set of all functions
ϕ∈ C(X)with compact supportS(ϕ), and letB^{+}_{b} (X)be the set of all positive
bounded Borel measurable functions on X.

For every µ ∈ M(X) let S(µ) denote the support of µ. Further, let ≺ denote (W-)specic order onM(X), that is,µ≺νif and only ifµ(u)≤ν(u)for all u∈ W. Trivially,µ≺ν, if µ≤ν. We recall that, for everyf:X→[0,∞],

R_{f} := inf{v∈ W:v≥f}
and, in particular, for allA⊂X andu∈ W,

R^{A}_{u} :=R_{1}_{A}_{u}= inf{v∈ W:v≥u onA}.

Moreover, Rˆ^{A}_{u}(x) := lim infy→xR^{A}_{u}, x ∈ X (so that Rˆ^{A}_{u} is the greatest
lower semicontinuous minorant of R^{A}_{u}). Let us note that, by the minimum
principle (see [3], III.6.6), for every y∈X and all neighborhoodsA of y,

(2.1) R^{A}_{G(·,y)}=G(·, y).

For everyx∈XandA⊂X, letε^{A}_{x} denote the measure obtained reducing
the Dirac measure ε_{x} onA, that ist,

Z

udε^{A}_{x} =R^{A}_{u}(x) (u∈ W).

Of course, ε^{A}_{x} ≺ ε_{x}, and ε^{A}_{x} = ε_{x} if x ∈ A. For every Borel measurable
set AinX and every µ∈ M(X), let

µ^{A}:=

Z

ε^{A}_{x} dµ(x)

(so that µ^{A} ≺ µ). The harmonic kernels HU for open sets U in X are then
obtained by H_{U}(x,·) :=ε^{X\U}x ,x∈X.

Given an open set W inX and a Borel measurable functionv≥0on X,
the function v is harmonic on W (superharmonic on W, respectively) pro-
vided, for every relatively compact open set U such thatU ⊂W, the function
H_{U}v is nite and continuous on W and H_{U}v = v (H_{U}v ≤ v, respectively).

Moreover, in order to avoid pathological situations, we shall tacitly assume that all superharmonic functions we consider are nite at nely isolated points (which other authors did without mentioning it explicitly). The superharmonic support C(v)of a superharmonic functionv≥0onX is the smallest closed set inX such thatv is harmonic on its complement.

If v≥0is superharmonic on X and (U_{n})is a sequence of relatively com-
pact open sets in X such thatU_{n}↑X, then the sequence (H_{U}_{n}v) is decreasing
and its inmum is the largest harmonic minorant hof v onX. The function v
is a potential on X if h = 0. LetP denote the set of all nite and continuous
potentials onX. It is known thatP is the set of allp∈ W ∩C(X)such thatp/v
vanishes at innity for some strictly positivev∈ W ∩ C(X)(see [3], Section II.5
and III.2.8).

As in ([3], Section III.5), letX_{0}denote the set of all points inXsuch that

(2.2) lim

U↓xH_{U}(x,·) =ε_{x}

(whereU runs over all open neighborhoods ofx). Then X\X0 is the set of all nely isolated points,

X\X_{0} ={x∈X:ε^{X\{x}}_{x} 6=ε_{x}},

and X\X_{0} is (at most) countable (see [3], III.7.2). If X is a discrete space,
then, of course, X_{0} =∅.

If y ∈X\X_{0}, then p_{y} := R^{{y}}_{1} ∈ P. If y is an isolated point in X, then
G(·, y)∈ P, and hence, G(·, y) =G(y, y)py by the minimum principle (see [3],
III.6.6); in particular, G(y, y)>0.

Let v ≥ 0 be superharmonic on X. Since, for every relatively compact
open U in X, the function H_{U}v is nite and continuous on U, the set P :=

{v=∞} ∩X_{0} has no interior points. Moreover,R_{1}^{P} = 0onX\P (consider the
functions (1/n)v ∈ W, n ∈ N) and hence, Rˆ^{P}_{1} = 0, that is, according to the
denition in ([3], Section VI.5),P is polar proving thatP has no nely interior
points (see [3], III.5.9). Havingv <∞onX\X0(due to our convention above),
we see that the set{v=∞}is polar.

For µ, ν∈ M(X), let us recall the denition (1.1) and introduce^{∗}G^{ν} by
G^{µ}:=

Z

G(·, y)dµ(y) and ^{∗}G^{ν} :=

Z

G(x,·)dν(x).

By Fatou's lemma, the functions G^{µ} and ^{∗}G^{ν} are lower semicontinuous.

Moreover, by Fubini's theorem,G^{µ}∈ W and

(2.3) Z

G^{µ}dν =
Z

∗G^{ν}dµ.

We shall need the following result (which is well known for harmonic spaces). For the convenience of the reader we include its proof.

Lemma 2.1. Let µ∈ M(X) such that G^{µ} is a potential. Then the super-
harmonic support of G^{µ} is the support of µ.

Proof. By Fubini's theorem, for every relatively compact open setU such
thatU does not intersectS(µ),HUG^{µ}=G^{µ}. So G^{µ}is harmonic on X\S(µ).
In particular, the superharmonic supportC of G^{µ} is contained inS(µ).

To prove the reverse inclusion, letKbe an arbitrary compact set inX\C
and ν := 1_{K}µ. Then G^{ν} is harmonic on X \K. Moreover, G^{ν} is harmonic
on X\C, sinceG^{ν} +G^{1}^{Kc}^{µ}=G^{µ} and G^{µ} is harmonic onX\C. So, by ([3],
III.4.5),G^{ν} is harmonic on X, and hence, G^{ν} = 0,ν = 0. Thus,µ(X\C) = 0,
that is,S(µ)⊂C.

Finally, we note the following.

Lemma 2.2. Let µ∈ M(X) such that G^{µ}∈ C(X). Then G^{µ}∈ P.

Proof. We know that v :=G^{µ}∈ W ∩ C(X). IfU is a relatively compact
open set in X, then H_{U}v = R^{U}_{v}^{c} is harmonic on U, by ([3], VI.2.6). So v is
superharmonic on X.

Let (U_{n}) be a sequence of relatively compact open sets in X such that
U_{n}↑X. By Fubini's theorem, for every n∈N,

H_{U}_{n}v=
Z

H_{U}_{n}G(·, y)dµ(y).

For everyy∈X, the functionG(·, y)is a potential, and hence,HUnG(·, y)↓0
asn→ ∞. Thus,H_{U}_{n}v↓0showing that vis a potential.

3. LIMITS OF REPRESENTABLE POTENTIALS

Lemma 3.1. Suppose that ν ∈ M(X) is supported by a compact set K
in X. Then ^{∗}G^{ν} is nite and continuous on X\K.

Proof. Let (yn) be a sequence in X\K which converges to a pointy0 in
X\K. By (i), the functionGis bounded onK× {y_{n}:n= 0,1,2, . . .}. So, by
(iii) and Lebesgue's theorem, limn→∞∗G^{ν}(yn) =^{∗}G^{ν}(y0)<∞.

Lemma 3.2. Let ϕ∈ K(X) with 0≤ϕ≤1, x∈X\S(ϕ), and ν:=

Z 1 0

ε^{{ϕ≥t}}_{x} dt.

Then ν is supported by S(ϕ), ν≺εx, and ^{∗}G^{ν} ∈ C(X). Moreover, ν does
not charge polar sets (semipolar sets, if (X,W) is a harmonic space).

Proof. Each measure ν_{t} := ε^{{ϕ≥t}}x is supported by {ϕ≥t} and ν_{t} ≺ε_{x}.
Therefore, ν is supported by S(ϕ) and ν ≺εx. By Lemma 3.1, ^{∗}G^{ν} is nite
and continuous atx. So lety0, y1, y2, . . . ∈X\ {x}with limn→∞yn=y0. For
all n∈N,

∗G^{ν}(yn) =
Z 1

0

∗G^{ν}^{t}(yn)dt,

where^{∗}G^{ν}^{t}(y_{n})≤G(x, y_{n}) andsup{G(x, y_{n}) : n= 0,1,2, . . .}<∞ by (i). Let
t ∈ (0,1). If ϕ(y0) < t, then limn→∞∗G^{ν}^{t}(yn) = ^{∗}G^{ν}^{t}(y0) by Lemma 3.1.

If ϕ(y0) > t, then we know, by (2.1), that ^{∗}G^{ν}^{t}(yn) = R^{{ϕ≥t}}_{G(·,y}

n)(x) = G(x, yn)
for n= 0 and all suciently largen, and hence, limn→∞∗G^{ν}^{t}(yn) = ^{∗}G^{ν}^{t}(y0)
by (iii). Thus, by Lebesgue's theorem,limn→∞∗G^{ν}(yn) =^{∗}G^{ν}(y0).

Moreover, the measures ν_{t}, 0 < t ≤ 1, do not charge (Borel) polar sets
(see [3], VI.5.6). So ν does not charge polar sets.

Suppose now that(X,W) is a harmonic space (see [3], p. 129) or [6]) and
let A be a semipolar Borel set in X. Then each measure ν_{t}, 0 < t ≤ 1, is
supported by the boundary of the closed set {ϕ≥t} (see [3], VI.2.12,2 or [6],
Proposition 7.1.3), which is a subset of{ϕ=t}. Since the level sets{ϕ=t}are
pairwise disjoint, we see, by ([9], Corollary 1.6), that all intersectionsA∩{ϕ=t}

except at most countably many are polar. Thus, ν(A) = 0.

Corollary 3.3. Let x∈X_{0} and u, v∈ W such that u(x)< v(x). Then
there exists ν ∈ M(X) which has compact support, does not charge (Borel )
polar sets (semipolar sets if (X,W) is a harmonic space) and satises ν ≺εx,

∗G^{ν} ∈ C(X), andν(u)< ν(v).

Proof (cf. [17]). Let (K_{n}) be a sequence of compact sets inX such that
K_{n} ↑ X\ {x}. Then R^{K}_{v}^{n} ↑ R^{X\{x}}v = v, since X\ {x} is nely dense in X.
So there exists n∈ N, such that u(x) < R^{K}_{v}^{n}(x). We may choose ϕ∈ K(X)
such that 0 ≤ ϕ≤1, ϕ = 1on Kn and ϕ = 0in a neighborhood of x. Then
K_{n}⊂ {ϕ≥t}for everyt∈(0,1). Deningν as in Lemma 3.2 we hence obtain
thatν(u)≤u(x)< R^{K}_{v}^{n}(x)≤ν(v).

Lemma 3.4. Let τ be a measure on a compact set K in X and let L be
a compact neighborhood ofK. Then^{∗}G^{τ}^{X\L}=^{∗}G^{τ} on X\L.

Proof. Consider y ∈ V := X\L. By (2.1), R^{V}_{G(·,y)} = G(·, y), that is,
ε^{V}_{x}(G(·, y)) =G(x, y) for everyx∈X. In particular,

∗G^{τ}^{V}(y) =τ^{V}(G(·, y)) =
Z

ε^{V}_{x}(G(·, y))dτ(x) =
Z

G(x, y)dτ(x) =^{∗}G^{τ}(y).

Proof of Theorem 1.1. If µ, ν ∈ M(X) such that p = G^{µ} = G^{ν}, then
µ=ν (see the Appendix, where this is shown under more general assumptions
on G).

Let us rst assume that(G^{µ}^{n}) converges top outside a polar set. Then
D:={x∈X:q(x)<∞, lim

n→∞G^{µ}^{n}(x) =p(x)}.

is polar. For the moment, let us x y∈X. Ify is isolated, then lim sup

n→∞ G(y, y)µn({y})≤lim sup

n→∞ G^{µ}^{n}(y)≤q(y)<∞,

where we know that G(y, y) > 0. Let us suppose now that y is not isolated.

Then, by (ii) and the density of D, there exists x_{y} ∈D such that x_{y} 6=y and
G(xy, y) > 0. By (iii), there exists an open neighborhood V of y and γ > 0
such thatG(xy,·)≥γ on V. Then, for everyn∈N,

γµn(V)≤ Z

G(xy, y^{0})dµn(y^{0})≤q(xy)<∞.

Hence every subsequence of (µ_{n})has a subsequence which is weakly con-
verging. Let (νn) be a subsequence of (µn) which converges weakly to µ ∈
M(X).

We intend to show that that G^{µ} = p. Due to the lower semicontinuity
of the functions G(x,·),

G^{µ}=
Z

G(·, y)dµ(y)≤ lim

n→∞

Z

G(·, y)dνn(y) =p≤q <∞ onD.

To prove the reverse inequality, let us suppose that there existsx∈Dsuch
thatG^{µ}(x)< p(x). By Corollary 3.3, there exists a measureν ∈ M(X) which
is supported by a compact set K in X, does not charge X\D and satises
ν ≺ ε_{x}, ^{∗}G^{ν} ∈ C(X), and ν(G^{µ}) < ν(p) (take ν = ε_{x} if x ∈ X\X_{0}). Let
0< δ < ν(p)−ν(G^{µ}). By ([3], III.6.1), there exists a compact neighborhoodL
of K inX such thatR^{X\L}p < δ/ν(K) on K. Let

(3.1) σ :=ν^{X\L}.

Then σ(p) = R

R^{X}p ^{\L}dν < δ. Moreover, σ(X\D) = 0 by ([3], VI.5.6).

The function ^{∗}G^{σ} is lower semicontinuous and, by Lemma 3.4, ^{∗}G^{σ} = ^{∗}G^{ν}
on X\L. Thus, the function^{∗}G^{ν}−^{∗}G^{σ} is upper semicontinuous and vanishes
on X\L. Therefore,

(3.2) lim

n→∞ν_{n}(^{∗}G^{ν} −^{∗}G^{σ})≤µ(^{∗}G^{ν}−^{∗}G^{σ}).

Since σ(X\D) =ν(X\D) = 0, we know that

n→∞lim ν(G^{ν}^{n}) =ν(p)<∞, lim

n→∞σ(G^{ν}^{n}) =σ(p)<∞, σ(G^{µ})≤ν(G^{µ})<∞.

So, by (3.2) and (2.3), ν(p)−σ(p) = lim

n→∞(ν−σ)(G^{ν}^{n})≤(ν−σ)(G^{µ})≤ν(G^{µ}),
henceν(p)−ν(G^{µ})≤σ(p)< δcontradicting our choice of δ.

Thereforep=G^{µ}onD, andhenceG^{µ}=p.Becauseof the uniqueness of a re-
presenting measure forpthis implies that the sequence(µ_{n})itself converges toµ.
Finally, let us assume that (X,W) is a harmonic space and that the set,
where (G^{µ}^{n}) does not converge to p may be semipolar. Then we only know
that X\D is semipolar so that ν^{X\L} might charge X\D. Hence we replace
(3.1) by

σ:=

Z 1 0

ν^{{ϕ≥t}}dt,

whereϕ∈ C(X)satisfying0≤ϕ≤1,ϕ= 0onLandϕ= 1outside a compact
neighborhood L^{0} of L. We recall that S(ν)⊂ K and note that, fort ∈(0,1),
X\L^{0} ⊂ {ϕ≥t} ⊂X\L. Arguing as at the end of the proof of Lemma 3.2, we
obtain that σ does not charge semipolar sets, and therefore σ(X\D) = 0. It
is easily veried thatσ(p)< δas before and now^{∗}G^{σ} =^{∗}G^{ν} onX\L^{0}. Hence,
the proof can be nished as before.

4. APPLICATION

Theorem 4.1. The following statements are equivalent.

1. For everyϕ∈ K^{+}(X), there exists ν∈ M(X) such that R_{ϕ} =G^{ν}.

2. For every nite continuous potential p on X having compact superhar-
monic support, there exists µ∈ M(X) such that p=G^{µ}.

3. For every potential p on X, there existsµ∈ M(X) such that p=G^{µ}.
4. There existsν ∈ M(X)such thatG^{ν} is a nite continuous strict potential.

5. There exists ν ∈ M(X), charging every nely open V 6= ∅, such that
G^{ν} ∈ C(X).

Proof. If ϕ ∈ K(X), ϕ ≥ 0, then Rϕ ∈ P and C(Rϕ) ⊂ S(ϕ) (see [3], III.5.6 and III.6.8). Moreover, there are nite continuous potentials which are strict (see [3], I.1.5). So the implications (3)⇒(2) ⇒(1) and (3)⇒ (4)hold trivially.

By Theorem 1.1, (1) implies (3). Indeed, if pis any potential onX, there
is a sequence (ϕ_{n}) inK^{+}(X) such thatϕ_{n}↑p, and hence R_{ϕ}_{n} ↑p.

To prove the equivalence of (4) and (5) let us consider ν ∈ M(X) such
that G^{ν} ∈ C(X). By Lemma 2.2, G^{ν} ∈ P. It is obvious that the potential
kernel associated with G^{ν} is given by f 7→ G^{f ν}, f ∈ B^{+}_{b} (X) (see Lemma 5.1
and [3], II.6.17). Moreover, ν charges every nely open non-empty set if and
only if G^{ν} is strict (see [3], VI.8.2).

It remains to show that (4) implies (3). So let p be a potential on X
and let ν ∈ M(X) such that q := G^{ν} is a nite continuous strict potential.

Since, of course, q > 0, we may consider the balayage space (X,W)˜ , where W˜ := (1/q)W. Then p/q is a potential and 1 = q/q is a strict potential with respect to (X,W˜). We dene

G(x, y) := (1/q(x))G(x, y)˜ (x, y∈X).

Then G˜ satises our assumptions and, for every µ ∈ M(X), G˜^{µ} = (1/q)G^{µ}.
In particular, G˜^{ν} = 1. There exists a sub-Markov resolvent V= (V_{λ})_{λ>0} onX
such thatW˜ is the set of all V-excessive functions and the potential kernel of V
is the potential kernel associated with1 (see [3], II.7.8). By ([3], II.3.11), there
exists a sequence (fn) in B_{b}^{+}(X) such that G˜^{f}^{n}^{ν} ↑p/q. Clearly, G˜^{f}^{n}^{ν} ∈ C(X),
n∈N. Thus, an application of Theorem 1.1 yields a measureµ∈ M(X) such
thatp/q= ˜G^{µ}, that is,p=G^{µ}.

5. APPENDIX: UNIQUENESS

In this section, we shall only need that G: X ×X → [0,∞] is Borel measurable and has the following property:

(ii^{0}) For everyy∈X,G(·, y)∈ W \ {0}and G(·, y) is harmonic onX\ {y}.
Let us observe that under this weak hypothesis it is not excluded that G(·, y)
is harmonic onX (which, of course, means that such aG(·, y)is useless for the
representation of potentials).

Lemma 5.1. Let µ ∈ M(X). Then G^{µ} ∈ W. Moreover, if G^{µ} is super-
harmonic on X\S(µ), then G^{µ} is harmonic on X\S(µ).

Proof. By Fatou's lemma, G^{µ} is lower semicontinuous. Let U be a rela-
tively compact open set in X. We know by (ii^{0}) that HUG(·, y) ≤G(·, y) for
everyy∈X. Hence, by Fubini's theorem,H_{U}G^{µ}≤G^{µ}. ThereforeG^{µ}∈ W.

Next, we suppose thatG^{µ}is superharmonic onX\S(µ)andU∩S(µ) =∅.
Then H_{U}G^{µ} is nite and continuous on U. Moreover, by Fubini's theorem,
H_{U}G^{µ}=G^{µ}, since, by (ii^{0}),H_{U}G(·, y) =G(·, y) for every y∈X\S(µ).

The proof of the uniqueness of representations is a slight modication of the proof given in ([1], p. 44) for Brelot spaces (cf. also [8], Theoreme 7, [12], Proposition 6.1 and [20], Lemma 2.5 for harmonic spaces, and [13], Proposi- tion 1.3 for balayage spaces).

Proposition 5.2. For every potential p on X, there exists at most one
measure µ∈ M(X) such that p=G^{µ}.

Proof. Let us assume that µ, ν ∈ M(X) with p = G^{µ} = G^{ν}, and let
D:={p <∞}.

Letµ˜:= (µ−ν)^{+}andν˜:= (ν−µ)^{+} so thatµ˜∧ν˜= 0. Sinceµ= ˜µ+µ∧ν
andν = ˜ν+µ∧ν we see thatG^{µ}^{˜} =G^{˜}^{ν} onD, and hence, onX. By Lemma 5.1,
G^{µ}^{˜} ∈ W so thatG^{µ}^{˜} is a potential. In other words, to prove thatµ=ν we may
assume without loss of generality that µ∧ν= 0.

Then there exist µn, νn ∈ M(X) such that µn ↑µ, νn ↑ ν, and S(µn)∩
S(ν_{n}) =∅ for every n∈N. For the moment we x n∈N and dene

fn:= 1_{{G}ν−νn<∞}(G^{µ}^{n}−G^{ν−ν}^{n})^{+}, gn:= 1_{{G}µ−µn<∞}(G^{ν}^{n}−G^{µ−µ}^{n})^{+}.
Then R_{f}_{n}, Rgn∈ W and there exist functions un, vn∈ W such that
(5.1) Rfn+un=G^{µ}^{n} and Rgn+vn=G^{ν}^{n}

(see [3], bottom of p. 43 and II.4.4). In particular, pn := Rfn is a potential
which is harmonic onX\S(µ_{n}), andqn:=Rgn is a potential which is harmonic
on X\S(ν_{n}). Since p_{n} is nely continuous, g_{n} is nely lower semicontinuous,
and p_{n} ≥ f_{n} = g_{n} on D, we see that p_{n} ≥ g_{n} on X, and hence, p_{n} ≥ q_{n}.
Similarly, qn ≥ pn. Thus, pn = qn, which implies that pn is harmonic on
(X\S(µ_{n}))∪(X\S(ν_{n})) =X (see [3], III.4.5). So p_{n} = 0. Consequently, by
the denition of p_{n},G^{µ}^{n} ≤G^{ν−ν}^{n} on D.

Letting n tend to innity, we conclude that G^{µ} ≤0 on D. Hence, G^{ν} =
G^{µ}= 0 onX. Thus, µ= 0 and ν= 0.

Acknowledgments. Ivan Netuka's research is supported in part by CRC-701, Bielefeld.

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Received 8 September 2013 Universitat Bielefeld, Fakultat fur Mathematik,

33501 Bielefeld, Germany

hansen@math.uni-bielefeld.de Charles University, Faculty of Mathematics and Physics,

Mathematical Institute, Sokolovska 83, 186 75 Praha 8,

Czech Republic netuka@karlin.m.cuni.cz