Conjugates of integral functionals on continuous functions

0.18 MB·

by  Ari-Pekka Perkkiö

Conjugates of integral functionals on continuous
7 functions
1
0
2 Ari-Pekka Perkki¨o∗

n
a
January 16, 2017
J

3
1

] Abstract
A
This article gives necessary and sufficient conditions for the dual rep-
F
resentation of Rockafellar in [Integrals which are convex functionals. II,
.
h PacificJ.Math.,39:439-469, 1971]forintegralfunctionalsonthespaceof
t continuous functions.
a
m
Keywords. integral functional; convex conjugate; set-valued mapping; inner
[ semicontinuity

1
v
1 Introduction
5
4
7 Let T be a σ-compact locally compact Hausdorff (σ-lcH) with the topology τ,
3 the Borel-σ-algebra B(T) and a strictly positive regular measure µ. The space
0 of Rd-valued continuous functions C := C(T;Rd), that vanish at infinity, is a
.
1 Banach space with respect to the supremum norm. The Banach dual of C can
0 be identified with the space M ofRd-valuedsignedregularmeasureswith finite
7
total variation on T via the bilinear form
1
:
v hy,θi:= ydθ;
i
X Z
r see [Fol99, Section 7].
a
This article studies conjugates of convex integral functionals on C. Such
functionalsappearinnumerousapplicationsinoptimalcontrol,variationalprob-
lems over functions of bounded variation, optimal transport, financial mathe-
matics,andinextremalvaluetheory. Applicationstosingularstochasticcontrol
and finance are given in [PP15].
Ourmaintheoremssharpenthoseof[Roc71]and[Per14]bygivingnecessary
and sufficient conditions for the conjugacy of integral functionals on C and
functionals on the space of measures with integral representation involving the
∗Department of Mathematics, Ludwig-Maximilians-Universit¨at Mu¨nchen, Theresienstr.
39, 80333 Mu¨nchen, Germany. The author is grateful to the V¨ais¨al¨a Foundation for the
financialsupport.
1
recessionfunctionoftheconjugateintegrand. Forcloselyrelatedresults,werefer
to [AB88, BV88] and the references therein. Our characterizations are new in
the literature and they complete the idea initiated in [Roc71] on expressing the
singular parts of the conjugates on the space of measures in terms of recession
functions.
Thepaperstartsbysharpening[Roc71,Theorem6]ontheintegralrepresen-
tation of the support function of continuous selections of a convex-valuedmap-
ping. We show that inner semicontinuity of the mapping is not only sufficient
but also a necessary condition for the validity of the representation. Section 3
gives the main results on integral functionals. The conditions appearing in the
main results are analyzed further in Section 4.
2 Inner semicontinuity and continuous selections
LetS beaset-valuedmappingfromT toRd anddenotethesetofitscontinuous
selections by
C(S):={y ∈C |y ∈S ∀t∈T}.
t t
This section sharpens [Roc71, Theorem 6] on the integral representationof the
support function
σ (θ):= sup hy,θi
C(S)
y∈C(S)
of C(S). Theorem 6 of [Roc71] states that if S is “inner semicontinuous” and
T is compact, then
σ (θ)= σ (dθ/d|θ|)d|θ| := σ ((dθ/d|θ|) )d|θ| ,
C(S) S St t t
Z Z
where|θ|denotesthetotalvariationofθandσ isthenormalintegranddefined
St
pointwise by
σ (x):= sup x·y.
St
y∈St
Theorem 1 below extends the result to general σ-lcH T and shows that in-
ner semicontinuity of S is a necessary condition for the validity of the integral
representation.
The proofof sufficiency is analogousof that of Rockafellar,but we repeat it
here for completeness since we do not assume compactness of T. To the best of
our knowledge, the necessity of the inner semicontinuity has not been recorded
in the literature before.
We denote the neighborhoods of t ∈ T by H and the open neighborhoods
t
of x ∈ Rd by Ho. The mapping S is inner semicontinuous (isc) at t if S ⊆
x t
liminfS , where
t
liminfS :={x∈Rd |S−1(A)∈H ∀A∈Ho}.
t t x
WhenS isiscateveryt,S isinnersemicontinuous. By[Mic56,Proposition2.1],
S isiscifandonlyifthepreimageS−1(O):={t∈T |S ∩O 6=∅}ofeveryopen
t
2
set O is open. Inner semicontinuity is studied in detail in [RW98, Section 5]
in the case where T is Euclidean. We often use Michael’s continuous selection
theorem[Mic56, Theorem3.1”’]thatis validfor aLindelo¨fperfectly normalT -
1
space T. We refer to [Dug66] for the definitions, and note here that Hausdorff
spacesareT andσ-compactnessofT impliesthateverysubsetofT isLindelo¨f,
1
so T is a Lindelo¨f perfectly normal T -space by [Dug66, Theorem XI.6.4 and
1
Problem VIII.6.7].
Given a set A, N (y) denotes the set of its normals at y, that is, θ ∈N (y)
A A
if y ∈A and
hθ,y′−yi≤0 ∀y′ ∈A,
while N (y) := ∅ for y ∈/ A. For a set-valued mapping S : T ⇒ Rd and a
A
function y :T →Rd, we denote the set-valued mapping t7→N (y ) by N (y).
St t S
The mapping t7→clS is called the image closure of S.
t
Theorem 1. Assume that S is a convex-valued mapping from T to Rd such
that C(S)6=∅. Then
σ (y)= σ (dθ/d|θ|)d|θ|
C(S) S
Z
if and only if S is inner semicontinuous. In this case θ ∈N (y) if and only
C(S)
if dθ/d|θ|∈N (y) |θ|-a.e.
S
Proof. We first prove sufficiency. The mapping defined by
Γ :=cl{y |y ∈C(S)}
t t
is contained in S, and, since a mapping is isc if and only if its image closure
is isc, Γ is isc by part 7 of Theorem 2 below. Assuming that S is not inner
semicontinuous,thereistsuchthatΓ isstrictlysmallerthanS . Byaseparat-
t t
ing hyperplane theorem, there is x with sup x·v < sup x·v. Defining
v∈Γt v∈St
θ ∈M so that |θ| is a Dirac measure at t and dθ/d|θ|=x, we can write this as
sup hy,θi< σ (dθ/d|θ|)d|θ|,
S
y∈C(S) Z
which is a contradiction.
Now we turn to necessity. By the Fenchel inequality,
hy,θi≤ σ (dθ/d|θ|)d|θ| (1)
S
Z
for every y ∈C(S), so it suffices to show
sup hy,θi≥ σ (dθ/d|θ|)d|θ|.
S
y∈C(S) Z
3
Denoting L∞(S)={w ∈L∞(T,θ)|w ∈S ∀t}, we have, by [RW98, Theorem
t t
14.60],
sup wdθ = σ (dθ/d|θ|)d|θ|.
S
w∈L∞(θ;S)Z Z
Let y¯∈C(D),
α< σ (dθ/d|θ|)d|θ|
S
Z
andw ∈L∞(S)besuchthat wdθ >α. ByregularityofθandLusin’stheorem
[Fol99,Theorem7.10],thereisanopenO¯ ⊂T suchthat (|y¯|+|w |)d|θ|<ǫ/2,
O¯C is compact and w is contRinuous relative to O¯C. The mO¯appingt
R
w if t∈O¯C
t
Γ =
t
(St if t∈O
is isc convex closed nonempty-valued so that, by [Mic56, Theorem 3.1”’], there
is a yw ∈ C with yw = w on O¯C and yw ∈ S for all t. Since |θ| is regular,
t t t t
there is an open Oˆ ⊃O¯C such that |yw|d|θ|<ǫ/2. Since O¯C is compact
Oˆ\O¯C t
and T is locally compact, we may choose Oˆ precompact, by [Dug66, Theorem
R
XI.6.2].
Since Oˆ and O¯ form an open cover of T and since T is normal, there is, by
[Mun00, Theorem 36.1], a continuous partition of unity (αˆ,α¯) subordinate to
(Oˆ,O¯). Defining y :=αˆyw+α¯y¯, we have y ∈C(S) and
ydθ ≥ wdθ− αˆ|yw|d|θ|− α¯|y¯|d|θ|≥ α−ǫ,
Z ZO¯C ZOˆ\O¯C ZO¯ Z
which finishes the proof of necessity, since α< σ (dθ/d|θ|)d|θ| was arbitrary.
S
As to the normal vectors, θ ∈ N (y) if and only if σ (y) = hθ,yi
C(D) C(D)
R
which, by the Fenchel inequality (1), is equivalent to
σ (dθ/d|θ|)d|θ| = y·(dθ/d|θ|)d|θ|
S
Z
which is equivalent to condition in the statement, since we also have the point-
wise Fenchel inequalities
σ (x)≥y ·x ∀x∈Rd
St t
that are satisfied as equalities if and only if x∈N (y ).
St t
The next theorem lists some useful criteria for the preservation of inner
semicontinuity in operations. Given a set A, rintA denotes its relative interior.
Theorem 2. If S,S1 and S2 are isc mappings, then
1. AS is isc for any continuous A:Rn →Rm,
4
2. coS is isc,
3. S1×S2 is isc,
4. S1+S2 is isc,
5. t7→S1∩S2 is isc provided G:={(t,x)|x∈S2} is open.
t t t
6. t7→S1∩S2 isiscprovidedT isEuclideanandS1 andS2 areconvex-valued
t t
with 0∈int(S1−S2) for all t,
t t
7. Arbitrary pointwise unions of isc mappings are isc.
Proof. 1 follows from (AS)−1(O) = S−1(A−1(O)), since then preimages of
open sets are open. 3 can be derived directly from the definition. Choosing
A(v1,v2)=v1+v2 and S =S1×S2, 4 follows from 1 and 3.
To prove 5, let A ⊂ Rd be open, t ∈ (S1∩S2)−1(A) and x ∈ S1∩S2∩A.
t t
ThesetG∩(T×A)isopenandcontains(t,x), sothereisA′ ∈Ho andO ∈Ho
x t
such that A′ ⊂A and O×A′ ⊂G. Then O∩(S1)−1(A′)⊆(S1∩S2)−1(A), so
(S1∩S2)−1(A)∈H . Thus (S1∩S2)−1(A) is open.
t
To prove 6, let tν →t. By [RW98, Theorem 4.32],
liminf(S1 ∩S2 )⊇liminfS1 ∩liminfS2 .
tν tν tν tν
Taking the intersection over all sequences tν → t proves the claim, by [RW98,
Exercise 5.6].
To prove 2, notice that coS is an union of all the mappings of the form
m αiS for m αi = 1, αi ≥ 0, so the result follows from 1,4 and 7. It
i=1 i=1
remains to prove 7. For any mappings (Sα), ( Sα)−1(A) = ((Sα)−1(A)).
P P α
Thus arbitrary unions of isc mappings is isc, since preimages of open sets are
S S
open.
3 Integral functionals of continuous functions
This section studies conjugates of functionals of the form
I +δ :C →R,
h C(D)
where h is a convex normal B(T)-integrandon Rd,
I (v):= h (v )dµ
h t t t
Z
and D := cldomh . Here domg := {x | g(x) < ∞} is the domain of g
t t
and cl denotes the closure operation. Recall that h is a convex normal B(T)-
integrand on Rd if its epigraphical mapping t 7→ epih := {(v,α) ∈ Rd ×R |
t
h (v) ≤ α} is closed convex-valued and measurable. A set-valued mapping S
t
is measurable if preimages of open sets are measurable. We refer the reader to
5
[RW98, Chapter 14]for a generalstudy of measurableset-valuedmappings and
normal integrands.
Note that, when h (v) = δ , we have I +δ = δ so we are in
t St(v) h C(D) C(D)
the setting of Section 2. On the other hand, when domI ⊆ C(D), we have
h
I +δ = I . Sufficient conditions for the above inclusion can be found in
h C(D) h
[Roc71], [Per14] and Section 4 below.
Recall that the conjugate of an extended real-valuedfunction F on C is the
extended real-valued lower semicontinuous (lsc) convex function on M defined
by
F∗(θ):= sup{hy,θi−F(y)}.
y∈C
Rockafellar [Roc71] and more recently Perkkio¨ [Per14] gave conditions for the
validity of the integral representation
(Ih)∗ =Jh∗,
where the functional Jh∗ :M →R is defined by
Jh∗(θ)= h∗t((dθa/dµ)t)dµt+ (h∗t)∞((dθs/d|θs|)t)d|θs|t,
Z Z
whereθa andθs denotetheabsolutelycontinuousandthesingularpart,respec-
tively, of θ with respect to µ, and (h∗)∞ denotes the recession function of h∗.
t t
Recall that the recession function of a proper lsc convex function g is given by
g(αx+x¯)−g(x¯)
g∞(x)=sup ,
α
α>0
where x¯∈domg is arbitrary;see [Roc70, Chapter 8]. We often use the identity
g∞ =σdomg∗ (2)
validforproperlscconvexfunctionsonlocallyconvextopologicalvectorspaces;
see [Roc66, Corollary 3D] or [Roc70, Theorem 13.3] for the finite dimensional
case. The identity (2) makes it clear that the properties of Jh∗ are related to
those of t7→D .
t
The following lemma is the first part of the proof of [Per14, Theorem 3]. It
shows that the functional Ih+δC(D) arises naturally as the conjugate of Jh∗.
Lemma 3. If domJh∗ 6=∅, then Jh∗∗ =Ih+δC(D).
6
Proof. We have
Jh∗∗(y)= sup ydθ−Jh∗(θ)
θ∈M(cid:26)Z (cid:27)
= sup y·θ′dµ−Ih∗(θ′)
θ′∈L1(µ;Rd)(cid:26)Z (cid:27)
+ sup y·wd|θs|− (h∗)∞(w)d|θs|
θ∈M,w∈L1(|θs|;Rd)(cid:26)Z Z (cid:27)
I (y) if y ∈cldomh ∀t,
h t t
=
(+∞ otherwise.
Here the second equality follows from the positive homogeneity of (h∗)∞, and
t
the third follows by first applying [RW98, Theorem 14.60] on the second and
the third line, where one uses (2) and then takes the supremum over all purely
atomic finite measures which are singular with respect to µ.
The next lemma combines [Roc71, Theorem 4] with the condition C(D) =
cl(domI ∩C(D)). This condition is part of the sufficient conditions in the
h
theorembelow,whichwillbereducedtothesituationinthe lemma. Wedenote
the closed ball centered at v with radius r by B (v) and the interior of a set A
r
by intA. When v =0, we write simply B .
r
Lemma 4. Assume that D is isc, C(D) = cl(domI ∩C(D)) and that y ∈
h
domI and r > 0 are such that t 7→ h (y + v) is finite and belongs to L1
h t t
whenever v ∈Br. Then (Ih+δC(D))∗ =Jh∗.
Proof. By [Roc71, Theorem 4], y ∈intdomI ∩C(D) and
h
Ih∗(θ)=mθi′n{Ih∗(dθ′/dµ)+σdomIh(θ−θ′)|θ′ ≪µ}.
Since intdomI ∩C(D) 6= ∅, we may apply Fenchel duality [Roc74, Theorem
h
20] to the sum I +δ . Thus
h C(D)
(I +δ )∗(θ)
h C(D)
=mθ′i′n{mθi′n{Ih∗(dθ′/dµ)+σdomIh(θ−θ′−θ′′)|θ′ ≪µ}+σC(D)(θ′′)}
=mθi′nmθ′i′n{Ih∗(dθ′/dµ)+σdomIh(θ−θ′−θ′′)+σC(D)(θ′′)|θ′ ≪µ}
=mθi′n{Ih∗(dθ′/dµ)+σdomIh∩C(D)(θ−θ′)|θ′ ≪µ}
=mθi′n{Ih∗(dθ′/dµ)+JσD(θ−θ′)|θ′ ≪µ}
=min{ h∗(dθ′/dµ)dµ+ (h∗)∞(d(θ−θ′)/dµ)dµ}
θ′
Z Z
+ (h∗)∞(d(θs)/d|θs|)d|θs|.
Z
7
Here the third equality follows from another application of Fenchel duality, the
fourth from the assumption C(D) = cl(domI ∩C(D)) and from Theorem 1.
h
By [Roc70, Corollary 8.5.1], the last minimum is attained at dθ′/dµ = dθ/dµ,
so the last expression equals Jh∗(θ).
The following theorem and its corollary are the main results of the paper.
The theoremgives necessaryandsufficient conditions for the conjugacy of I +
h
δC(D) and Jh∗. The corollarysharpens [Roc71, Theorem5] by giving necessary
and sufficient conditions for the conjugacy between Ih and Jh∗.
Given a function g, ∂g(y) denotes its subgradients at y, that is, θ ∈∂g(y) if
g(y)+hy′−y,θi≤g(y′) ∀y′.
Given a normal integrand h and y ∈ C, we denote the set-valued mapping
t7→∂(h )(y ) by ∂h(y) and the mapping t7→N (y ) by ∂sh(y).
t t Dt t
Theorem 5. Assuming Ih+δC(D) and Jh∗ are proper, they are conjugates of
each other if and only if domh is isc and C(D)=cl(domI ∩C(D)), and then
h
θ ∈∂(I +δ )(y) if and only if
h C(D)
dθa/dµ∈∂h(y) µ-a.e.,
(3)
dθs/d|θs|∈∂sh(y) |θs|-a.e.
Proof. Toprovethe necessity,assumethatIh+δC(D) andJh∗ areconjugatesof
eachother. By(2),σ =J∞,where,bythemonotoneconvergence
cldom(Ih+δC(D)) h∗
theorem, Jh∞∗ = J(h∗)∞. Here, by (2) again, (h∗)∞ = σD. By Lemma 3,
J∗ = δ , so cl(domI ∩C(D)) = C(D). In particular, σ =J , so D
σD C(D) h C(D) σD
is inner semicontinuous, by Theorem 1, which shows the necessity.
Toprovethesufficiency,itsuffices,byLemma3andthebiconjugatetheorem
[Roc74, Theorem 5], to show that Jh∗ is lsc. For ǫ>0, we define
ǫh (x):= min h (x+x′)
t t
x′∈Bǫ
and ǫD := cldomǫh . By [RW98, Proposition 14.47], ǫh is a convex normal
t t
integrand, and a direct calculation gives
ǫh∗(v)=h∗(v)+ǫ|v|.
t t
When ǫց0, both Iǫh+δC(Dǫ) րIh+δC(D) and Jǫh∗ ցJh∗ pointwise, so, Jh∗
is lsc by Lemma 9 in the appendix provided that (Iǫh +δC(Dǫ))∗ = J(ǫh)∗. To
have this, we will apply Lemma 4 to ǫh.
We have ǫD = cldomh +B , so ǫD is isc, by Theorem 2. To show that
t ǫ
C(ǫD)=cl(domIǫh∩C(ǫD)), let y ∈C(ǫD). For any ν =1,2,..., the mapping
t7→Dt∩intBǫ(1+2−ν−1)(yt) is convex nonempty-valued and isc, by Theorem2.
By [Mic56, Theorem 3.1”’] and the assumption C(D) = cl(domI ∩ C(D)),
h
there is y˜ν ∈ domI ∩C(D) such that ky−y˜νk ≤ ǫ(1+2−ν). Defining yν =
h
(1−ǫ2−ν)y+ǫ2−νyν,wehaveyν ∈C(ǫD)andyν →yinC. Sincekyν−y˜νk<ǫ,
8
we get Iǫh(yν) ≤ Ih(y˜ν) and yν ∈ domIǫh. Thus y ∈ cl(domIǫh ∩ C(ǫD).
Finally, since Ih and Jh∗ are proper, we have, for any y¯∈domIh∩C(D), that
t 7→ ǫh (y¯ +v) is finite and integrable whenever v ∈ B . Thus ǫh satisfies the
t t ǫ
assumptions of Lemma 4 which finishes the proof the sufficiency.
As to the subdifferential formulas, we have, for any y ∈ dom(I +δ )
h cldomh
and θ ∈M, the Fenchel inequalities
h(y)+h∗(dθa/dµ)≥y·(dθa/dµ) µ-a.e.,
(h∗)∞(dθs/d|θs|)≥y·(dθs/d|θs|) |θs|-a.e.
(thelatterholdsby(2)),soθ ∈∂Ih(v)ifandonlyifIh(y)+Jh∗(θ)=hy,θiwhich
is equivalent to having the Fenchel inequalities satisfied as equalities which in
turn is equivalent to the given pointwise subdifferential conditions.
Remark 1. The properness of Ih+δC(D) and Jh∗ is equivalent to the existence
of x¯∈L1(Rd), y¯∈C(D) and α∈L1 such that
h (v)≥v·x¯ −α ,
t t t
h∗(x)≥y¯ ·x−α .
t t t
Indeed, for y ∈ domIh∩C(D) and θ ∈ Jh∗, we may choose y¯= y, x¯ = θ/dµ,
and α := max{h∗(x¯),h(v¯)}. Choosing y = y¯ and θ by dθ/dµ = x¯ and θs = 0,
gives the other direction.
Just like in Theorem 1, the necessity of inner semicontinuity in Theorem 5
has not been explicitly stated in the literature before. The domain condition
will be analyzed more in detail in the next section.
The followingcorollaryextends [Roc71,Theorem5]and[Per14,Theorem2]
inthecaseoflcHσ-compactT. ItbuildsonTheorem5bygivingnecessaryand
sufficient conditions for the equality I +δ =I .
h C(D) h
Corollary 6. Assuming Ih and Jh∗ are proper, they are conjugates of each
other if and only if domh is isc and C(D) = cldomI , and then θ ∈ ∂I (y) if
h h
and only if
dθa/dµ∈∂h(y) µ-a.e.,
dθs/d|θs|∈∂sh(y) |θs|-a.e.
Proof. Since C(D) = cldomI implies that I = I +δ and that C(D) =
h h h C(D)
cl(domI ∩C(D)), we have sufficiency by Theorem 5. Assuming that I and
h h
Jh∗ are conjugates of each other, Lemma 3 implies that Ih +δC(D) = Ih, so
domI ∩C(D)=domI , and the sufficiency follows from Theorem 5.
h h
TherelatedmaintheoremsbyRockafellarandPerkkio¨in[Roc71]and[Per14]
give sufficient conditions for Corollary6. Rockafellar’s assumptioninvolvedthe
notion of ”full lower semicontinuity” of set-valued mappings whereas Perkkio¨
formulatedtheassumptionintermsof”outerµ-regularity”. We returntothese
concepts in the next section.
9
4 The domain conditions in the main results
In this section we will analyze more in detail the conditions
C(D)=cl(domI ∩C(D)),
h
C(D)=cldomI
h
appearing in Theorem 5 and Corollary 6. The former is characterizedin Theo-
rem 7 below whereas for the latter we give sufficient conditions in terms of the
domain mapping D.
Theconditioninthenextresult,whichextendstheconditionin[Roc71,The-
orem5],canbefoundin[BV88,Proposition6]formetrizableσ-lcHT. Herewe
generalizeto σ-lcH T and give the conversestatement for inner semicontinuous
domains.
Theorem 7. Let Ih+δC(D) and Jh∗ be proper. If, for every t ∈ T and every
v ∈rintD ,thereisO ∈H andy ∈C(D)suchthaty =v and h(y)dµ<∞,
t t t O
then
R
C(D)=cl(domI ∩C(D)).
h
The converse holds if D is isc.
Proof. Lety0 ∈domI ∩C(D), ǫ>0andy ∈C(D). ThereisacompactK ⊂T
h
such that ky0k,kyk ≤ ǫ/2 outside K. By compactness and the assumptions,
thereexistt ∈T,i=1,...,nforwhichthereareyi ∈C(D)andO ∈H such
i ti ti
that |yi−y| ≤ ǫ on O , h(yi)dµ < ∞ and K ⊂ n O . Let α0,αi, i =
ti Oti i=1 ti
1,...,n be continuous partition of unity subordinate to {KC,O ,...O }. By
convexity, n αiyi ∈ (dRomI ∩C(D)), and, by conSstruction,tk1 tαniyti −
i=0 h i=1
yk≤ǫ. Since ǫ was arbitrary, we see that y ∈cl(domI ∩C(D)).
h
P P
Assume now that D is isc and that C(D) = cl(domI ∩C(D)). Let t ∈ T
h
andv ∈rintD . Thereexistsvi ∈rintD ,i=0,...,d,suchthatv ∈co{vi |i=
t t
0,...d}. Let ǫ > 0 be small enough so that v ∈ co{v¯i | i = 0,...d} whenever,
for all i, v¯i ∈Bvi,ǫ. By Michael selectiontheorem[Mic56, Theorem3.1”’],there
is,for everyi, acontinuousselectionofD takinga valuevi att, so,thereexists
yi ∈domIh∩C(D) suchthat yti ∈Bvi,ǫ. Letαi ∈(0,1)be convexweightssuch
that d αiyi = v. By convexity, d αiyi ∈ C(D)∩domI which proves
i=0 t i=0 h
the claim.
P P
We now turn to the condition C(D) = cldomI in Corollary 6. We say
h
that a function y : T → Rd is a µ-selection of D if y ∈ D outside a µ-null
t t
set. Clearly, every y ∈ domI is a continuous µ-selection of D. Thus, if every
h
continuous µ-selection of D is a selection of D, we have domI ⊆ C(D) and
h
thus, I +δ =I . If, in adddition, C(D)=cl(domI ∩C(D)), we thus get
h C(D) h h
C(D)=cldomI .
h
Following [Per14], we say that a set-valued mapping S is outer µ-regular if
µ-liminfS ⊆clS ∀t∈T,
t t
10