Lower semicontinuity of monotone functionals in the mixed topology on Cb

Nendel, Max
A icle — Published Ve sion
Lowe semicon inui y o mono one unc ionals in he
mixed opology on Cb
Finance and S ochas ics
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Sugges ed Ci a ion: Nendel, Max (2024) : Lowe semicon inui y o mono one unc ionals in he mixed
opology on Cb, Finance and S ochas ics, ISSN 1432-1122, Sp inge , Be lin, Heidelbe g, Vol. 29, Iss.
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Finance and S ochas ics (2024) 29:261–287
h ps://doi.o g/10.1007/s00780-024-00552-2
Lowe semicon inui y o mono one unc ionals in he
mixed opology on Cb
Max Nendel1
Recei ed: 18 Oc obe 2022 / Accep ed: 23 July 2024 / Published online: 28 No embe 2024
© The Au ho (s) 2024
Abs ac
The main esul o his pape cha ac e ises he con inui y om below o mono one
unc ionals on he space Cbo bounded con inuous unc ions on an a bi a y Pol-
ish space as lowe semicon inui y in he mixed opology. In his pa icula si ua ion,
he mixed opology coincides wi h he Mackey opology o he dual pai (Cb,ca),
whe e ca deno es he space o all coun ably addi i e signed Bo el measu es o ini e
a ia ion. Hence lowe semicon inui y in he mixed opology is o con ex mono one
maps Cb→Requi alen o a dual ep esen a ion in e ms o coun ably addi i e
measu es. Such ep esen a ions a e o undamen al impo ance in inance, e.g. in he
con ex o isk measu es and supe hedging p oblems. Based on he main esul , eg-
ula i y p ope ies o capaci ies and dual ep esen a ions o Choque in eg als in e ms
o coun ably addi i e measu es o 2-al e na ing capaci ies a e s udied. Mo eo e , a
well-known cha ac e isa ion o s a -shaped isk measu es on L∞is ans e ed o isk
measu es on Cb. In a second s ep, he pape p o ides a cha ac e isa ion o equicon-
inui y in he mixed opology o amilies o con ex mono one maps. As a conse-
quence, o e e y con ex mono one map on Cb aking alues in a locally con ex
ec o la ice, con inui y in he mixed opology is equi alen o con inui y on no m-
bounded se s.
Keywo ds Risk measu e ·Mono one unc ional ·Choque in eg al ·Con inui y om
below ·Lowe semicon inui y ·Mixed opology ·Mackey opology ·S a -shaped
Ma hema ics Subjec Classifica ion 91G70 ·46A20 ·28A12
JEL Classifica ion C02 ·C65
1In oduc ion
In his pape , we s udy con inui y p ope ies o mono one maps Cb→R, whe e
Cb=Cb() deno es he space o all bounded con inuous unc ions on a Polish
M. Nendel
[email p o ec ed]
1Cen e o Ma hema ical Economics, Biele eld Uni e si y, Uni e si ä ss aße 25, 33615
Biele eld, Ge many
262 M. Nendel
space wi h alues in R. Mono one unc ionals Cb→Rappea in many
applica ions. Special ins ances o such maps, in he con ex o inance and ac ua ial
science, a e
– isk measu es o nonlinea expec a ions; c . Denis e al. [11], Föllme and
Schied [15, Chap. 4] and Peng [22, Chap. 1].
– supe hedging unc ionals; c . Che idi o e al. [7] and Che idi o e al. [8].
– obus expec ed u ili ies o loss unc ions; c . Delbaen [9,10].
– Choque in eg als, e.g. in he con ex o insu ance p emia; c . Wang e al. [25].
In o de o ob ain dual ep esen a ions o con ex mono one unc ionals in e ms
o coun ably addi i e measu es, addi ional con inui y p ope ies a e usually equi ed.
The wo mos p ominen con inui y p ope ies in his con ex a e con inui y om
abo e and con inui y om below; c . Föllme and Schied [15, Lemma 4.21 and The-
o em 4.22]. Fo con ex mono one unc ionals, con inui y om below is usually a
weake equi emen han con inui y om abo e; see o ins ance Che idi o e al. [8].
In he ield o ma hema ical inance, hese con inui y p ope ies ha e been s udied
in many con ex s in he pas decades. Fo isk measu es, con inui y om abo e is (up
o a di e en sign con en ion) ela ed o he Lebesgue p ope y, whe eas con inui y
om below is closely ied o he Fa ou p ope y. The Lebesgue and Fa ou p ope ies
e e o sequen ial con inui y and lowe semicon inui y, espec i ely, o uni o mly
bounded poin wise con e gen sequences o measu able unc ions, and Fa ou closed-
ness is a undamen al ing edien in no-a bi age heo y; see e.g. Bu zoni and Mag-
gis [5] and He degen and Khan [18]. Fixing a e e ence measu e and wo king on
L∞, i is well known ha con inui y om below o con ex isk measu es is equi a-
len o a dual ep esen a ion in e ms o coun ably addi i e measu es; c . Föllme and
Schied [15, Theo em 4.33]. I he isk measu e is also law-in a ian , con inui y om
below is au oma ically sa is ied i he unde lying p obabili y space is assumed o be
a omless; c . Jouini e al. [19].
On he o he hand, mone a y isk measu es a e closely linked o nonlinea ex-
pec a ions and he opic o model unce ain y in inance. In his con ex , isk mea-
su es which a e no domina ed by a single p obabili y measu e ha deems e en s o
be negligible o no play a c ucial ole. An example o such a isk measu e is he
G-expec a ion; c . Denis e al. [11] and Peng [22, Chap. 2]. Howe e , on he space
Bbo bounded measu able unc ions wi hou a e e ence p obabili y, con inui y om
below alone is no su icien o gua an ee a dual ep esen a ion o con ex mono one
unc ionals in e ms o coun ably addi i e measu es, despi e he ac ha i implies
sequen ial lowe semicon inui y o such unc ionals in he weak opology σ(Bb,ca)
o he dual pai (Bb,ca). In Denk e al. [12, Example 3.6], an example is gi en o a
cohe en isk measu e which is con inuous om below on Bb, bu does no ha e a sin-
gle coun ably addi i e mino an . On he o he hand, con inui y om abo e o a isk
measu e on he space o bounded measu able unc ions al eady implies he exis ence
o a domina ing e e ence measu e; see e.g. Denk e al. [12, Rema k 3.3].
One way ou o his dilemma is o es ic a en ion o con inuous claims. Fo he
space Cbo bounded con inuous unc ions as an unde lying unc ion space, i is well
known ha con inui y om abo e, o a con ex mono one unc ional Cb→R,is
su icien bu no necessa y o a dual ep esen a ion in e ms o coun ably addi i e
Lowe semicon inui y o mono one unc ionals on Cb263
measu es; see e.g. Che idi o e al. [8]. Howe e , he ques ion whe he such a ep-
esen a ion is equi alen o he weake no ion o con inui y om below on gene al
Polish s a e spaces has emained unanswe ed o almos a decade, as discussed in he
in oduc ion o Delbaen [10]. In a se ies o pape s, his ques ion has been answe ed
posi i ely by Delbaen [9,10], and as a consequence, con ex mono one unc ionals on
Cbwhich a e con inuous om below a e lowe semicon inuous in he Mackey opol-
ogy μ(Cb,ca)o he dual pai (Cb,ca), whe e ca deno es he space o coun ably ad-
di i e signed Bo el measu es wi h ini e a ia ion. F om a ma hema ical pe spec i e,
his is a ema kable esul , since he Mackey opology μ(Cb,ca)is no me isable and
con inui y om below is a equi emen o sequences, so ha nonme isabili y poses
a majo p oblem. The e o e in [9,10], ano he pa h is chosen and he p oo s he e
ely on compac i ica ion me hods; mo e p ecisely, hey use he ac ha e e y Polish
space can be embedded as a Gδin o a compac me ic space.
Theo em 2.2 in he p esen pape gene alises he main esul o Delbaen [9]by
showing ha o any mono one unc ional Cb→R, con inui y om below is equi -
alen o lowe semicon inui y in he mixed opology. The la e is a classical gen-
e al concep in analysis, c . Wiwege [27], and coincides wi h he Mackey opology
μ(Cb,ca)in his pa icula se ing. Mo eo e , Theo em 2.2 shows ha o mono-
one unc ionals Cb→R, uppe o lowe semicon inui y in he mixed opology a e
equi alen o sequen ial uppe o lowe semicon inui y in he mixed opology, espec-
i ely, despi e he ac ha he mixed opology is no me isable. Since he Mackey
opology μ(Cb,ca)is he ines opology leading o he dual space ca o coun ably
addi i e signed Bo el measu es wi h ini e a ia ion, i is a na u al choice o duali y
heo y on Cb. In pa icula , Theo em 2.2 implies ha e e y con ex mono one unc-
ional on Cbadmi s a dual ep esen a ion in e ms o coun ably addi i e measu es;
c . Co olla y 2.5. Howe e , using he explici ep esen a ion o a local base a he
o igin o he mixed opology allows us o u he cha ac e ise con inui y om below
in e ms o p oximi y on compac se s also o noncon ex mono one unc ionals; see
Theo em 2.2.
In Co olla y 2.6, we u n ou ocus o capaci ies and Choque in eg als. In a i s
s ep, we cha ac e ise he egula i y o gene al capaci ies, de ined on open se s, in
e ms o con inui y om below o he Choque in eg al on he se Lbo all bounded
lowe semicon inuous unc ions →Rand o he capaci y along sequences o open
se s. In a second s ep, we cha ac e ise 2-al e na ing capaci ies, o which he ela ed
Choque in eg al admi s a dual ep esen a ion in e ms o coun ably addi i e measu es
on Lb, in e ms o con inui y om below along sequences o open se s and egula i y
o he capaci y, pa ially ex ending he esul s in Adamski [1].
Co olla y 2.7 ex ends a well-known cha ac e isa ion o s a -shaped isk measu es
on L∞, discussed in Cas agnoli e al. [6], o gene al isk measu es on he space
Cb. While he p oo ollows closely ha in [6], Co olla y 2.5 allows he ansi ion
om L∞ o Cb.
Ano he ques ion we add ess in his pape is a cha ac e isa ion o con inui y o
con ex mono one maps in he mixed opology. Con inui y in he mixed opology
is o undamen al impo ance in many si ua ions in obus inance. In he con ex o
supe hedging, i has been s udied in Che idi o e al. [7]. In he con ex o dynamic isk
measu es and semig oups ela ed o s ochas ic p ocesses unde model unce ain y, i
264 M. Nendel
appea s in Blessing e al. [3], Goldys e al. [17] and wi h a di e en language, i is also
used in he analysis o la ge de ia ion p inciples based on max-s able isk measu es;
c . Kuppe and Zapa a [21]. Building on Theo em 2.2, we discuss he equicon inui y
o amilies o con ex mono one maps in he mixed opology in Theo em 2.8. The e,
a cha ac e isa ion o equicon inui y in e ms o con inui y om abo e and uni o m
equicon inui y on sup emum-no m-bounded se s is gi en.
F om a ma hema ical pe spec i e, he mixed opology has wo s iking ea u es.
On he one hand, unless is compac , i has no neighbo hood o ze o which is
bounded wi h espec o he sup emum no m. On he o he hand, i has he in in-
sic p ope y ha o linea ope a o s aking alues in an a bi a y locally con ex
space, con inui y is equi alen o con inui y on sup emum-no m-bounded se s. Co ol-
la y 2.11, which is a consequence o Theo em 2.8, ex ends his in insic p ope y by
showing ha o con ex mono one maps aking alues in a locally con ex ec o la -
ice, μ(Cb,ca)-con inui y is equi alen o μ(Cb,ca)-con inui y on sup emum-no m-
bounded se s. A pa icula ins ance o a locally con ex ec o la ice is Cbi sel , en-
dowed wi h he mixed opology, which in a inancial con ex co esponds o example
o he case o condi ional isk measu es o condi ional nonlinea expec a ions.
The es o he pape is o ganised as ollows. In Sec . 2, we s a e he main esul s
and hei co olla ies. Sec ion 3con ains all p oo s. In Appendix A, we p o e an aux-
ilia y esul o capaci ies and Choque in eg als, and in Appendix B, we p o e an
auxilia y esul on locally con ex ec o la ices.
2 Main esul s
Th oughou ,le be a Polish space and Cb=Cb() he space o all bounded
con inuous unc ions →R. We conside he local base
V2:= {{g∈Cb:g∞< }: >0}
a 0 ∈Cb o he opology induced by he sup emum no m ·∞, and he local base
V1:=g∈Cb:sup
x∈C|g(x)|<
: >0,C⊆compac 
a 0 ∈Cb o he ec o opology o uni o m con e gence on compac s. Le Vdeno e
he sys em consis ing o all se s

n∈N
n
∑
k=1
(V 1
k∩kV 2)wi h (V 1
k)k∈N⊆V1and V2∈V2,(2.1)
whe e kV2:= {kg :g∈V2} o all k∈Nand
n
∑
k=1
Vk:= n
∑
k=1
gk:g1∈V1,...,g
n∈Vn

Lowe semicon inui y o mono one unc ionals on Cb265
o n∈Nand nonemp y subse s V1,...,V
no Cb. Then Vis a local base a 0 ∈Cb
o a Hausdo locally con ex opology β, which is known as he mixed opology.We
e e o Wiwege [27] o a de ailed discussion on he mixed opology in a mo e gen-
e al se ing. Clea ly, he mixed opology βis ine han he weak opology σ(Cb,ca)
o he dual pai (Cb,ca), whe e ca deno es he space o all coun ably addi i e signed
Bo el measu es o ini e a ia ion. A well-known ac , which we do no make use o ,
is ha he mixed opology βcoincides wi h he Mackey opology o he dual pai
(Cb,ca). Mo eo e , βbelongs o he class o s ic opologies; c . Wheele [26]. We
also e e o F emlin e al. [16] and Sen illes [24] o addi ional ine p ope ies o
mixed o s ic opologies.
We say ha a unc ional U:Cb→Ris mono one i U( ) ≤U(g) o all
, g ∈Cbwi h ≤g, whe e o unc ions →R, he ela ion ≤and all o he
o de - ela ed objec s e e o he poin wise o de .
Fo a sequence ( n)n∈N⊆Cband a unc ion :→R, we w i e n
as n→∞i n≤ n+1 o all n∈Nand (x) =limn→∞ n(x) o all x∈.
Analogously, we w i e n as n→∞i n≥ n+1 o all n∈Nand
(x)=limn→∞ n(x) o all x∈.
Defini ion 2.1 a) We say ha a mono one unc ional U:Cb→Ris con inuous om
below i U( ) =limn→∞ U( n) o all sequences ( n)n∈N⊆Cband ∈Cbwi h
n as n→∞.
b) We say ha a mono one unc ional U:Cb→Ris con inuous om abo e i
U( ) =limn→∞ U( n) o all sequences ( n)n∈N⊆Cband ∈Cbwi h n
as n→∞.
Theo em 2.2 Le U:Cb→Rbe mono one.Then he ollowing a e equi alen :
(i) Uis con inuous om below.
(ii) Uis lowe semicon inuous in he mixed opology β.
(iii) Uis sequen ially lowe semicon inuous in he mixed opology β.
(i ) Fo all ∈Cb,ε>0and ≥0, he e exis δ>0and a compac C⊆
such ha
U( ) ≤U( +e) +ε
o all e∈Cbwi h e∞≤ and supx∈C|e(x)|≤δ.
Fo an a bi a y unc ion U:Cb→R, i s conjuga e unc ion U:Cb→Ris gi en
by U( ) := −U(− ) o all ∈Cb. Then we ob ain he ollowing co olla y.
Co olla y 2.3 Le U:Cb→Rbe mono one.Then he ollowing a e equi alen :
(i) Uis con inuous om abo e.
(ii) Uis uppe semicon inuous in he mixed opology β.
(iii) Uis sequen ially uppe semicon inuous in he mixed opology β.
(i ) Fo all ∈Cb,ε>0and ≥0, he e exis δ>0and a compac C⊆
such ha
U( +e) ≤U( )+ε
o all e∈Cbwi h e∞≤ and supx∈C|e(x)|≤δ.
266 M. Nendel
A combina ion o Theo em 2.2 and Co olla y 2.3 leads o he ollowing cha ac e -
isa ion o con inui y in he mixed opology o mono one unc ionals.
Co olla y 2.4 Le U:Cb→Rbe mono one.Then he ollowing a e equi alen :
(i) Uis con inuous om abo e and below.
(ii) Uis con inuous in he mixed opology β.
(iii) Uis sequen ially con inuous in he mixed opology β.
(i ) Fo all ∈Cb,ε>0and ≥0, he e exis δ>0and a compac C⊆
such ha
|U( +e) −U( )|≤ε
o all e∈Cbwi h e∞≤ and supx∈C|e(x)|≤δ.
As a di ec consequence o Theo em 2.2, we ob ain he main esul in Delbaen [9].
We deno e by ca+ he se o all posi i e elemen s o ca, i.e., he se o all ini e Bo el
measu es.
Co olla y 2.5 Le U:Cb→Rbe con ex and mono one.Then he ollowing a e
equi alen :
(i) Uis con inuous om below.
(ii) Uis lowe semicon inuous in he mixed opology β.
(iii) Uis lowe semicon inuous in he weak opology σ(Cb,ca).
(i ) The e exis a nonemp y se M⊆ca+and a unc ion α:M→Rsuch ha
U( ) =sup
μ∈M(
dμ−α(μ)) o all ∈Cb.(2.2)
We apply Co olla y 2.5 o he case o Choque in eg als. In he sequel, le Ode-
no e he amily o all open subse s o , i.e., he opology on , and Lb he se o
all bounded lowe semicon inuous unc ions →R.Acapaci y (on O)is a map
c:O→[0,∞)wi h
c(∅)=0 and c(B1)≤c(B2) o all B1,B
2∈Owi h B1⊆B2.
Fo a capaci y c:O→[0,∞), we de ine he Choque in eg al wi h espec o cas

dc:=∞
0c({ >s})ds+0
−∞ (c({ >s})−c())ds o all ∈Lb.
By de ini ion, he Choque in eg al is posi i ely homogeneous, i.e.,

(λ ) dc=λ
dc o all ∈Lband λ>0,
and cons an addi i e, i.e.,

( +m) dc=
dc+mc() o all ∈Lband m∈R.
Lowe semicon inui y o mono one unc ionals on Cb267
A well-known ac is ha he Choque in eg al is subaddi i e, i.e.,

( 1+ 2)dc≤
1dc+
2dc o all 1,
2∈Lb,
i and only i he capaci y cis 2-al e na ing, i.e.,
c(B1∪B2)+c(B1∩B2)≤c(B1)+c(B2) o all B1,B
2∈O.
Fo he eade ’s con enience, we p o ide a p oo o his s a emen in Appendix A.
Ano he consequence o Theo em 2.2 and Co olla y 2.5 is he ollowing esul
conce ning he egula i y o gene al capaci ies and dual ep esen a ions o Choque
in eg als in e ms o coun ably addi i e measu es o 2-al e na ing capaci ies. We
poin ou ha he equi alence be ween (i) and (ii) is a s anda d esul om Choque
heo y and can be ound o example in he ex book by König [20, Exe cise 11.18].
The equi alence be ween (iii) and (i ) has been discussed in a mo e gene al se ing in
Adamski [1]. The main no el y is he implica ion (ii) ⇒(iii), which is a consequence
o Theo em 2.2 and oge he wi h Co olla y 2.5 acili a es he p oo o he emaining
equi alences. Fo he eade ’s con enience, we p o ide a sel -con ained p oo o all
equi alences.
Co olla y 2.6 Le c:O→[0,∞)be a capaci y.Then he ollowing a e equi alen :
(i) Fo e e y sequence (Bn)n∈N⊆Owi h Bn⊆Bn+1 o all n∈N,
c(
n∈N
Bn)=lim
n→∞c(Bn).
(ii) The Choque in eg al is con inuous om below on Lb,i.e.,

dc=lim
n→∞
ndc
o all ∈Lband any sequence ( n)n∈N⊆Lbwi h n as n→∞.
(iii) The capaci y cis egula ,i.e., o all B∈O,
c(B) =sup
CB
in
A∈O
A⊇C
c(A), (2.3)
whe ewew i eCB o C⊆B⊆wi h Ccompac .
I cis 2-al e na ing, he s a emen s (i)–(iii) a e equi alen o he ollowing s a e-
men :
(i ) The e exis s a nonemp y se M⊆ca+wi h μ() =c() o all μ∈Mand

dc=sup
μ∈M
dμ o all ∈Lb.
We now p esen an applica ion o Co olla y 2.5 o s a -shaped isk measu es
on Cb. In he ollowing, we say ha a mono one unc ional R:Cb→Ris a isk
268 M. Nendel
measu e i R(0)=0 and R( +m) =R( ) +m o all ∈Cband all cons an s
m∈R. We say ha a isk measu e R:Cb→Ris s a -shaped i
R(λ ) ≤λR( ) o all ∈Cband λ∈[0,1].
Fo a de ailed discussion on isk measu es, we e e o Föllme and Schied [15,
Chap. 4], and o a su ey on s a -shaped isk measu es o Cas agnoli e al. [6]. The
ollowing co olla y is a a ian o [6, P oposi ion 5] in ou se ing. The p oo hea -
ily uses he insigh s ob ained in he p oo o [6, Theo em 2]. We poin ou ha [6,
P oposi ion 5] co e s only he case o domina ed isk measu es, i.e., isk measu es
on L∞, which is discussed in de ail in [15, Sec . 4.3], whe eas we conside gene al
isk measu es es ic ed o he space o bounded con inuous unc ions.
Co olla y 2.7 Le R:Cb→R.Then he ollowing a e equi alen :
(i) The map Ris a s a -shaped isk measu e.
(ii) The e exis a nonemp y se Iand a amily (αi)i∈Io unc ions ca1
+→[0,∞]
wi h in μ∈ca1
+αi(μ) =0 o all i∈Iand
R( ) =min
i∈Isup
μ∈ca1
+(
dμ−αi(μ)) o all ∈Cb.
We conclude his sec ion wi h a ious cha ac e isa ions o con inui y in he mixed
opology o con ex mono one maps. We s a wi h he ollowing heo em which is
ou second main esul . Recall V om (2.1).
Theo em 2.8 Le Ibe a nonemp y index se and (Ui)i∈Ia amily o con ex and
mono one maps Cb→Rwi h
sup
i∈I(Ui( ) −Ui(0))<∞ o all cons an s ≥0.(2.4)
Then he ollowing a e equi alen :
(i) Fo e e y sequence ( n)n∈N⊆Cbwi h n0as n→∞,
lim
n→∞sup
i∈I(Ui( n)−Ui(0))=0.
(ii) Fo e e y ≥0and e e y ε>0, he e exis s some V∈Vwi h
sup
 ∞≤
sup
i∈I|Ui( +e) −Ui( )|<ε o all e∈V.
(iii) Fo e e y ≥0and e e y ε>0, he e exis a compac C⊆and a cons an
M≥0such ha
sup
i∈I|Ui( 1)−Ui( 2)|≤Msup
x∈C| 1(x) − 2(x)|+ε
o all 1,
2∈Cbwi h max{ 1∞, 2∞}≤ .
Lowe semicon inui y o mono one unc ionals on Cb275
Temam [14, P oposi ion 3.1], (ii) is equi alen o he ac ha Uadmi s a dual
ep esen a ion o he o m
U( ) =sup
μ∈M(μ −α(μ)) o all ∈Cb
wi hase Mo β-con inuous linea unc ionals on Cband a unc ion α:M→R.
Le μ∈Mand ∈Cbwi h ≥0. Since Uis mono one, i ollows ha
1
λ(μ(−λ ) −α(μ))≤U(−λ )
λ≤U(0)
λ o all λ>0.
Hence
0=lim
λ→∞−U(0)+α(μ)
λ≤μ ,
which shows ha e e y linea unc ional in Mis posi i e. The emaining equi a-
lences and in pa icula he dual ep esen a ion (2.2) now ollow om he ac ha
by Theo em 2.2 and he Daniell–S one heo em, c . Bogache [4, Theo em 7.8.1], a
posi i e linea unc ional μ:Cb→Ris con inuous in he mixed opology βi and
only i i belongs o ca+.
P oo o Co olla y 2.6 1) We i s p o e he implica ion (i) ⇒(ii). To ha end, le
( n)n∈N⊆Lband ∈Lbwi h n as n→∞. Then o all s∈R,
lim
n→∞c({ n>s})=c({ >s}).
Using he mono one con e gence heo em, i ollows ha
lim
n→∞
ndc=lim
n→∞(∞
0c({ n>s})ds+0
−∞ (c({ n>s})−c())ds)
=∞
0c({ >s})ds+0
−∞ (c({ >s})−c())ds
=
dc. (3.7)
2) Fo he implica ion (ii) ⇒(iii), i s obse e ha
c(B) ≥sup
CB
in
A∈O
A⊇C
c(A) o all B∈O.
In o de o show he con e se inequali y, le B∈Oand ε>0. In a i s s ep, we
conside he case B=. Then by Theo em 2.2, he e exis δ>0 and a compac se
C⊆such ha

1dc≤
gdc+ε

276 M. Nendel
o all g∈Cbwi h g∞≤1 and supx∈C|g(x) −1|≤δ.Nowle A∈Owi h
A⊇C. Since Ais open, he e exis s some m∈Nsuch ha g:→Rgi en by
g(x) := sup
y∈C(1−md(x, y))∨0 o all x∈
sa is ies g(x) =0 o x∈ A. Since 0 ≤g≤1 and g(x) =1 o all x∈C,i
ollows ha
c() =
1dc≤
gdc+ε≤
1Adc+ε=c(A) +ε.
We ha e he e o e shown ha
c() ≤in
A∈O
A⊇C
c(A) +ε
Taking he sup emum o e all Cand le ing ε→0 yields (2.3).
Fo gene al B∈O, he s a emen now ollows om he ac ha B, endowed wi h
he subspace opology
OB:={A∩B:A∈O}={A∈O:A⊆B}⊆O,
is again a Polish space, oge he wi h he obse a ion ha a subse o Bis compac in
he subspace opology OBi and only i i is compac in he o iginal opology O.
3) To p o e ha (iii) implies (i), le (Bn)n∈N⊆Owi h Bn⊆Bn+1 o all n∈N
and ε>0. Then he e exis s some compac C⊆n∈NBn=:Bwi h
c(B) ≤in
A∈O
A⊇C
c(A) +ε.
Since Cis compac , C⊆B=n∈NBnand Bnis open wi h Bn⊆Bn+1 o all
n∈N, he e exis s some n0∈Nsuch ha C⊆Bn0. Hence
in
A∈O
A⊇C
c(A) ≤c(Bn0),
and i ollows ha
sup
n∈Nc(Bn)≤c(B) ≤sup
n∈Nc(Bn)+ε=lim
n→∞c(Bn)+ε.
Le ing ε→0, i ollows ha c(B) =limn→∞ c(Bn).
4) Now we assume ha he capaci y cis 2-al e na ing. In o de o p o e ha (i )
implies (i), le (Bn)n∈N⊆Owi h Bn⊆Bn+1 o all n∈N. Then
c(
n∈N
Bn)=sup
μ∈M
μ(
n∈N
Bn)
=sup
μ∈M
sup
n∈Nμ(Bn)=sup
n∈Nsup
μ∈M
μ(Bn)=lim
n→∞c(Bn).
Lowe semicon inui y o mono one unc ionals on Cb277
5) In he las s ep, we p o e ha (ii) implies (i ). By Co olla y 2.5, he e exis a
se M⊆ca+and a unc ion α:M→Rsuch ha

dc=sup
μ∈M(
dμ−α(μ)) o all ∈Cb.
Since he Choque in eg al is posi i ely homogeneous, i ollows ha

dμ−α(μ)
λ=1
λ(
λ dμ−α(μ))≤1
λ
λ dc=
dc
o all ∈Cb,μ∈Mand λ>0. Le ing λ→∞, i ollows ha
sup
μ∈M
dμ≤
dc o all ∈Cb.
On he o he hand,
−α(μ) =
0dμ−α(μ) ≤
0dc=0 o all μ∈M.
Hence o all ∈Cb,
sup
μ∈M
dμ≤
dc=sup
μ∈M(
dμ−α(μ))≤sup
μ∈M
dμ.
In pa icula ,
μ() =
1dμ≤
1dc=c() o all μ∈M.
Since he Choque in eg al is cons an addi i e, i ollows ha
0=c() +
(−1)dc≥c() +
(−1)dμ=c() −μ()
o all μ∈M. We ha e he e o e shown ha μ() =c() o all μ∈M. De ining
o ∈Lb,x∈and n∈N he quan i y n(x) := in y∈( (y) +nd(x, y)) wi h
ame icdconsis en wi h he opology on , he e exis s o all ∈Lba sequence
( n)n∈N⊆Cbwi h n as n→∞, and so

dc=sup
n∈N
ndc
=sup
n∈Nsup
μ∈M
ndμ=sup
μ∈M
sup
n∈N
ndμ=sup
μ∈M
dμ,
whe e he i s equali y uses (3.7) and he las he mono one con e gence heo em.

278 M. Nendel
P oo o Co olla y 2.7 Fi s assume ha (ii) is sa is ied, i.e., he e exis a se I=∅and
a amily (αi)i∈Io unc ions ca1
+→[0,∞] wi h in μ∈ca1
+αi(μ) =0 o all i∈I
and
R( ) =min
i∈Isup
μ∈ca1
+( dμ−αi(μ)) o all ∈Cb.
Then one eadily e i ies ha Ris mono one wi h R( +m) =R( ) +m o all
∈Cband m∈R. Since in μ∈ca1
+αi(μ) =0 o all i∈I, i ollows ha
R(0)=min
i∈Isup
μ∈ca1
+(−αi(μ))=min
i∈I(−in
μ∈ca1
+
αi(μ))=0.
We ha e he e o e shown ha Ris a isk measu e. To p o e ha Ris s a -shaped, le
∈Cband λ∈[0,1]. Then
R(λ ) =min
i∈Isup
μ∈ca1
+(
λ dμ−αi(μ))
≤min
i∈I(λsup
μ∈ca1
+
dμ−αi(μ))=λR( ).
To p o e he con e se implica ion (i) ⇒(ii), assume ha Ris a s a -shaped isk
measu e. Following he p oo o Cas agnoli e al. [6, Theo em 2], we de ine
Aϕ:= { ∈Cb:∃λ∈[0,1]wi h ≤λ(ϕ−R(ϕ))} o all ϕ∈Cb.
Then o all ϕ∈Cb, hese Aϕis con ex wi h g∈Aϕ o all g∈Cbwi h g≤
o some ∈Aϕ, and 0 ∈Aϕ,m/∈Aϕ o all m∈(0,∞). Indeed, o he
la e , assume owa ds a con adic ion ha he e exis m∈(0,∞)and λ∈(0,1]
wi h m≤λ(ϕ −R(ϕ)). Then ϕ≥m
λ+R(ϕ), which con adic s he ac ha Ris
a isk measu e. Hence by Föllme and Schied [15, P oposi ion 4.7], he unc ional
Rϕ:Cb→Rgi en by
Rϕ( ) := in {m∈R: −m∈Aϕ} o all ∈Cb
de ines a con ex isk measu e on Cb.Le ∈Cband m∈Rwi h
m>in ϕ∈CbRϕ( ). Then he e exis s some ϕ∈Cbwi h m>R
ϕ( ), and so
−m≤λ(ϕ−R(ϕ)) o some λ∈[0,1].
Since Ris a s a -shaped isk measu e, i ollows ha
R( ) −m=R( −m) ≤Rλ(ϕ−R(ϕ))≤λR(ϕ−R(ϕ))=0.
Hence R( ) ≤in ϕ∈CbRϕ( ) o all ∈Cb. Mo eo e , o all ϕ∈Cband m∈R
wi h m<R(ϕ), i ollows ha ϕ−m>ϕ−R(ϕ) so ha Rϕ(ϕ) =R(ϕ) by he
Lowe semicon inui y o mono one unc ionals on Cb279
de ini ion o Aϕ. Indeed, i ϕ−R(ϕ) ≤0, i ollows ha R(ϕ) =supx∈ϕ(x).In
his case, he inequali y m<R(ϕ)implies ha he e exis s some x∈wi h
ϕ(x) −m>0≥λ(ϕ(x) −R(ϕ)) o all λ∈[0,1].
We ha e he e o e shown ha
R( ) =min
ϕ∈Cb
Rϕ( ) o all ∈Cb.
In iew o Co olla y 2.5 and [15, Theo em 4.16], i emains o p o e ha
Rϕ:Cb→Ris con inuous om below o all ϕ∈Cb. To ha end, le ϕ∈Cb,
( n)n∈N⊆Cbwi h n ∈Cbas n→∞, and (mn)n∈N⊆Rwi h
mn>supk∈NRϕ( k) o all n∈Nand limn→∞ mn=supk∈NRϕ( k). Then he e
exis s a sequence (λn)n∈N⊆[0,1]such ha
n−mn≤λn(ϕ−R(ϕ)) o all n∈N.
Since [0,1]is compac , by passing o a subsequence, we may wi hou loss o
gene ali y assume ha λn→λ∈[0,1]as n→∞. Then
−sup
k∈NRϕ( k)=lim
n→∞( n−mn)≤lim
n→∞λn(ϕ−R(ϕ))=λ(ϕ −R(ϕ)),
which p o es ha Rϕ( ) ≤supk∈NRϕ( k). Since Rϕ:Cb→Ris mono one, i
ollows ha Rϕ( ) =limn→∞ Rϕ( n).
P oo o Theo em 2.8 Fi s obse e ha by con exi y o Ui o all i∈Iand (2.4),
sup
i∈I(Ui( 1)−Ui( 2))≤sup
i∈I(Ui( 1)+Ui(− 2)−2Ui(0))
≤2sup
i∈I(Ui( ) −Ui(0))<∞(3.8)
o all ≥0 and 1,
2∈Cbwi h max{ 1∞, 2∞}≤ .
1) The implica ion (iii) ⇒(i) ollows om Rema k 3.1 a). To p o e ha (i) implies
(ii), we i s show ha o e e y ∈Cband e e y ε>0, he e exis s some V∈V
wi h
sup
i∈I|Ui( +e) −Ui( )|<ε o all e∈V.
To ha end, le ∈Cband conside he mono one maps U ,U :Cb→Rgi en by
U (g) := sup
i∈I(Ui( +g) −Ui( )),
U (g) := sup
i∈I(Ui( ) −Ui( −g)) o all g∈Cb.
Obse e ha by (3.8), U and U a e well de ined and
U (−g) =sup
i∈I(Ui( ) −Ui( +g)) o all g∈Cb.
280 M. Nendel
Mo eo e , o any V∈V,e∈Vi and only i −e∈V. Hence he auxilia y s a emen
ollows om Co olla y 2.3 once we ha e shown ha bo h U and U a e con inuous
om abo e. To ha end, le g∈Cband (gn)n∈N⊆Cbwi h gngas n→∞.
Mo eo e , le ε>0, n∈Nand λ∈(0,1). Then using he ac ha Uiis con ex o
all i∈I,
U (gn)−U (g) ≤sup
i∈I(Ui( +gn)−Ui( +g))
≤λsup
i∈I(Uign−g
λ−Ui(0))+λsup
i∈I(Ui(0)−Ui( +g))
+(1−λ) sup
i∈I(Ui +g
1−λ−Ui( +g))
and
U (gn)−U (g) ≤sup
i∈I(Ui( −g) −Ui( −gn))
≤sup
i∈I(Ui( −2g+gn)−Ui( −g))
≤λsup
i∈I(Uign−g
λ−Ui(0))+λsup
i∈I(Ui(0)−Ui( −g))
+(1−λ) sup
i∈I(Ui −g
1−λ−Ui( −g)).
Since he maps
R→R,γ→ sup
i∈IUi(γ( ±g))−Ui( ±g)
a e con ex and he e o e con inuous, we ob ain
λsup
i∈I(Ui(0)−Ui( ±g))+(1−λ) sup
i∈I(Ui ±g
1−λ−Ui( ±g))<ε
2
o λ∈(0,1)su icien ly small. Mo eo e , by assump ion,
λsup
i∈I(Uign−g
λ−Ui(0))<ε
2
o n∈Nsu icien ly la ge since gn−g
λ0asn→∞. We ha e he e o e shown ha
0≤U (gn)−U (g) < ε and 0 ≤U (gn)−U (g) < ε
o n∈Nsu icien ly la ge, and so bo h U and U a e con inuous om abo e. We
ha e hus p o ed he auxilia y s a emen and a e now eady o p o e he implica ion
(i) ⇒(ii). Fo i∈Iand ∈Cb,le Ui, :Cb→Rbe gi en by
Ui, (g) := Ui( +g) −Ui( ) o all g∈Cb.

Lowe semicon inui y o mono one unc ionals on Cb281
Then Ui, is con ex and mono one wi h Ui, (0)=0 o all i∈Iand ∈Cb.
Mo eo e , o all λ∈(0,1),i∈Iand , g ∈Cb,
Ui, (g) ≤λ(Uig
λ−Ui( ))+(1−λ)(Ui
1−λ−Ui( ))
≤λ(Uig
λ−Ui(0))+λ(Ui(− )−Ui(0))
+(1−λ)(Ui
1−λ−Ui( )),
whe e he second inequali y uses he ac ha Ui(0)−Ui( ) ≤Ui(− )−Ui(0) o
all i∈I.Le ≥0, (gn)n∈N⊆Cbwi h gn0asn→∞and ε>0. Then by (3.8),
he map
R→R,γ→ sup
 ∞≤
sup
i∈I(Ui(γ ) −Ui( ))
is con ex and well de ined. The e o e i is con inuous and i ollows ha
λ(Ui(− )−Ui(0))+(1−λ)(Ui
1−λ−Ui( ))<ε
2
o λ∈(0,1)su icien ly small. Hence we ge
sup
 ∞≤
sup
i∈I
Ui, (gn)≤λsup
i∈I(Uign
λ−Ui(0))+ε
2<ε
o n∈Nsu icien ly la ge, and we ha e shown ha
lim
n→∞ sup
 ∞≤
sup
i∈I
Ui, (gn)=0.
Using he auxilia y s a emen o he con ex mono one unc ions Ui, wi h i∈Iand
∈Cbwi h  ∞≤ , he e exis s some V∈Vsuch ha
sup
 ∞≤
sup
i∈I|Ui( +e) −Ui( )|= sup
 ∞≤
sup
i∈I|Ui, (e)|≤ε o all e∈V.
2) I emains o p o e he implica ion (ii) ⇒(iii). Le ≥0 and ε>0. Then he e
exis s some V∈Vsuch ha
sup
 ∞≤
sup
i∈I|Ui( +e) −Ui( )|≤ε o all e∈V.
By he de ini ion (2.1) o he local base V, he e exis some compac C⊆and
some δ>0 such ha
e∈Cb:sup
x∈C|e(x)|<δ
∩{e∈Cb:e∞≤2 }⊆V.
282 M. Nendel
Now le 1,
2∈Cbwi h max{ 1∞, 2∞}≤ . Then by he iangle inequali y,
 1− 2∞≤2 .I sup
x∈C| 1(x) − 2(x)|<δ, hen
sup
i∈I|Ui( 1)−Ui( 2)|≤ε.
On he o he hand, i supx∈C| 1(x) − 2(x)|≥δ, hen by (3.8), i ollows ha
sup
i∈I|Ui( 1)−Ui( 2)|≤Msup
x∈C| 1(x) − 2(x)|
wi h M:= 2
δsupi∈I(Ui( ) −Ui(0)). The p oo is comple e. 
P oo o Co olla y 2.9 Since (L, τ) is a locally con ex ec o la ice, (i ) implies (i).
The implica ion (i) ⇒(iii) is i ial, and by Co olla y 2.3, (ii) and (iii) a e equi alen .
By Theo em 2.8 and Lemma B.1, (ii) ⇒(i ) and (ii) ⇒( ), since (ii) oge he wi h
Dini’s lemma implies ha o e e y con ex and weak∗compac se Ko nonnega i e
τ-con inuous linea unc ionals, he map
UK:Cb→R, → sup
μ∈K
μ(U( )−U(0))
is con ex wi h limn→∞ UK( n)=0 o e e y sequence ( n)n∈N⊆Cbwi h n0
as n→∞. Since o e e y nonnega i e τ-con inuous linea unc ional λ:L→R,
he e exis s a τ-con inuous la ice semino m p:L→[0,∞)wi h |λu|≤p(u) o
all u∈L, ( ) implies (ii) by Dini’s lemma. 
Appendix A: Capaci ies and Choque in eg als
The se up and no a ion in his sec ion ollow ha o he main pa . The ollowing
lemma is a so o olklo e esul ; c . König [20, P ope y 11.8 and Theo em 11.11].
Fo he eade ’s con enience, we ne e heless p o ide a sho p oo .
Lemma A.1 Le c:O→[0,∞)be a capaci y.Then he ollowing a e equi alen :
(i) Fo all B1,B
2∈O,
c(B1∪B2)+c(B1∩B2)≤c(B1)+c(B2).
(ii) Fo all 1,
2∈Lb,

( 1+ 2)dc≤
1dc+
2dc.
P oo We i s p o e he implica ion (ii) ⇒(i). To ha end, le B1,B
2∈O. Then
c(B1∪B2)+c(B1∩B2)=
(1B1+1B2)dc
≤
1B1dc+
1B2dc=c(B1)+c(B2).
Lowe semicon inui y o mono one unc ionals on Cb283
We p oceed wi h he p oo o he implica ion (i) ⇒(ii). In he i s s ep, we p o e by
induc ion o e n∈N ha

n
∑
i=1
1Bidc≤
n
∑
i=1
c(Bi) o all B1,...,B
n∈O.(A.1)
Fo n=1, he s a emen is i ial. Assume ha (A.1)isp o ed o somen∈Nand
le B1,...B
n+1∈O. Then
n+1
∑
k=1
1Bi=1n+1
i=1Bi+
n
∑
k=1
1(k
i=1Bi)∩Bk+1.
Using (A.1) and (i), we ob ain ha

n+1
∑
i=1
1Bidc=c(n+1

i=1
Bi)+
n
∑
k=1
1(k
i=1Bi)∩Bk+1dc
≤c(n+1

i=1
Bi)+
n
∑
k=1
c(k

i=1
Bi∩Bk+1)≤
n+1
∑
i=1
c(Bi).
Now le 1,
2∈Lb. Since he Choque in eg al is cons an addi i e, we may wi hou
loss o gene ali y assume ha 1≥0 and 2≥0. Le := max{ 1∞, 2∞}.Fo
i=1,2, n∈Nand k=1,...,2n, de ine Bk
i,n := { k>k2−n }. Then o i=1,2,
 i−
2n
∑
k=1
2−n 1Bk
i,n ∞≤2−n −→ 0asn→∞.
Bu posi i e homogenei y o he Choque in eg al and (A.1)gi e

2n
∑
k=1
2−n (1Bk
1,n +1Bk
2,n )dc=2−n 
2n
∑
k=1
(1Bk
1,n +1Bk
2,n )dc
≤2−n
2n
∑
k=1(c(Bk
1,n)+c(Bk
2,n))
=
2n
∑
k=1
2−n 1Bk
1,n dc+
2n
∑
k=1
2−n 1Bk
2,n dc,
and so i ollows ha

( 1+ 2)dc≤
1dc+
2dc. 
284 M. Nendel
Appendix B: Locally con ex ec o la ices
Thoughou his sec ion,le (L, τ) be a locally con ex ec o la ice, i.e., a ec o
la ice L oge he wi h a locally con ex opology τon Lwhich is gene a ed by a
amily o la ice semino ms.
Le L+:= {u∈L:u≥0}and Lbe he opological dual space o L, i.e., he
space o all con inuous linea unc ionals L→R. Mo eo e , le
L+:= {λ∈L:λu ≥0,∀u∈L+}
be he se o all posi i e con inuous linea unc ionals on L.Fo u∈L,weuse he
s anda d no a ion u+:= u∨0 and u−:= −(u ∧0). Then u=u+−u−and
|u|:=u++u− o all u∈L. The ollowing lemma can be deduced om Alip an is
and Bo de [2, Theo em 8.24 and Co olla y 8.25] oge he wi h he ac ha e e y
linea unc ional ha is bounded by a la ice semino m is o de bounded. Fo he sake
o a sel -con ained exposi ion, we p o ide a sho p oo .
Lemma B.1 Le p:L→[0,∞)be a con inuous la ice semino m.
a) Fo e e y λ∈Lwi h |λu|≤p(u) o all u∈L, he e exis λ+,λ
−∈L+wi h
λu =λ+u−λ−uand max{λ+|u|,λ
−|u|} ≤ p(u) o all u∈L. (B.1)
b) The e exis s a con ex and weak∗compac se K⊆L+wi h
max
μ∈K|μu|≤max
μ∈Kμ|u|=p(u) ≤2max
μ∈K|μu| o all u∈L.
P oo a) Since pis a la ice semino m, i ollows ha p(u) =p( ) o all u, ∈L
wi h |u|=| |. In pa icula ,
p(u) =p(|u|) o all u∈L. (B.2)
Le λ∈Lwi h |λu|≤p(u) o all u∈L, and de ine
λ+u:= sup{λ : ∈L+, ≤u} o all u∈L+.
Since pis a la ice semino m, 0 ≤λ+u≤p(u) and λ+(αu) =αλ+u o all
u∈L+and α≥0. In o de o p o e ha λ+is addi i e, le u1,u
2∈L+. Then
o 1,
2∈L+wi h 1≤u1and 2≤u2,
λ 1+λ 2≤λ( 1+ 2)≤λ+(u1+u2).
Hence λ+u1+λ+u2≤λ+(u1+u2). On he o he hand, o ∈L+wi h ≤u1+u2,
le
1:= ( −u2)+≥0,
2:= − 1= +(u2− ) ∧0= ∧u2≤u2.
Mo eo e , 1≤u1since u1≥0, and 2= ∧u2≥0 since ≥0 and u2≥0.
Hence
λ =λ 1+λ 2≤λ+u1+λ+u2.