On general characterization of Young measures associated with Borel functions

0.13 MB·

by  Andrzej Z. Grzybowski

On general characterization of Young measures associated with
Borel functions
Andrzej Z. Grzybowski
Instituteof Mathematics, Czestochowa University of Technology, al. Armii Krajowej 21,
42-200 Cz¸estochowa, Poland
6
1 Email: [email protected]
0 Piotr Pucha la
2 Instituteof Mathematics, Czestochowa University of Technology, al. Armii Krajowej 21,
42-200 Cz¸estochowa, Poland
n
a Email: [email protected], [email protected]
J
2 Abstract
We provethat the Youngmeasure associated with a Borel function f
]
A is a probability distribution of the random variable f(U), where U has a
uniform distribution on thedomain of f. As an auxiliary result, the fact
F ∗
thatYoungmeasuresassociated withsimplefunctionsareweak densein
.
h thesetofYoungmeasuresassociatedwithmeasurablefunctionsisproved.
t
a
m Keywords: quasi-Youngmeasures, Youngmeasures; uniform distribution
AMS Subject Classification: 46N30; 49M30; 60A10
[

1
v
1 Introduction
6
0
2 One of the major problems in the calculus of variations is minimization of
0
functionals which are bounded from below but do not attain their infima. If
0
the minimize functional J is bounded, the direct method canbe applied: there
.
1 alwaysexistsaminimizingsequenceforJ,thatisasequence(u ),u : Rd →Rl,
n n
0 n ∈ N, such that lim J(u ) = infJ. Additionally, if J is coercive, (u ) is
6 n→∞ n n
1 alwaysbounded. However,ifJ doesnotattainitsinfimumthentheelementsof
: (u )are functions ofhighly oscillatorynature. Moreover,weak∗ convergencein
v n
i L∞ of (un) to some function u0 does not guarantee, that the sequence (ϕ(un))
X
of compositions of u with continuous function ϕ is weak∗ convergent in L∞.
n
r Indeed, in general it is not convergent not only to ϕ(u ), but to any function
a 0
with domain in Rd.
LaurenceChisholmYoungintroducedin[11]objectscalledbyhim’generali-
zed curves’, nowadays called ’Young measures.’ These are the ’generalized li-
mits’of sequencesofhighly oscillating functions. The ’mature’formof Young’s
theorem has been proved by J.M.Ball in [3]. According to these theorems,
we say that that under their assumptions the considered sequences ’generate’
appropriate Young measures. This approach is studied for example in [8] in
detail.
Alternatively,wecanlookattheYoungmeasureasatobjectassociatedwith
any measurablefunctiondefinedonanonempty,open,boundedsubsetΩ ofRd
with values in a compact subset K of Rl. Such a conclusion can be derived
1
from the theorem 3.6.1 in [10]. Thank to this theorem it can be proved that
theYoungmeasureassociatedwithasimplefunctionisthe convexcombination
of Dirac measures. These Dirac measures are concentrated at the values of the
simplefunctionunderconsiderationwhilecoefficientsoftheconvexcombinaton
are proportional to the Lebesgue measure of the sets on which the respective
values are taken on by the function, see [9] for details and more general result.
In this article we significantly generalize the above results. We prove a
theoremproviding generalyet simple description of Young measuresassociated
with Borel functions. As a consequence, the theorem enables one to compute
explicitformulaeofprobabilitydensityfunctionsoftheYoungmeasuresinmany
interesting cases. This can be done without commonly used to calculate weak∗
limits of sequences of functions generating Young measures functional analytic
apparatus. Since Young measures are widely used in many areas of theoretical
andappliedsciences(seeforexample[2],[4],[5],[8]),ourresultprovideahandy
tool of obtaining their explicit form.
The main theorem of this article states that the Young measure associated
withanyBorelfunctionf definedonthesetΩ ⊂RdwithpositiveLebesguemea-
sureM andvaluesinacompactsetK ⊂Rl,isinfactaprobabilitydistribution
of a random variable X = f(U), where U is uniformly distributed on Ω. Be-
forethis,weprovealemmacorrespondingtostandardmeasure-theoreticresult,
thatanyBorelfunctionisapointwiselimitoftheappropriatesequenceofsimple
functions. Relyingonthisfactweprove,thatforanyBorelfunctionf: Ω →K,
itsYoungmeasureistheweak∗limitofasequenceofYoungmeasuresassociated
with the elements of the sequence of simple functions convergentpointwise to f.
2 Young measures – necessary information and
an auxiliary result
Thefirstpartofthis sectioncanserveasaverybriefintroductiontothe theory
of Young measures. In the second part we state and prove a lemma mentioned
at the end of the Introduction.
2.1 An outline of the Young measures theory
We gather now some information about Young measures. An interested reader
can find details, together with proofs and further bibliography, for example in
[1], [6], [7], [8], [10].
Let Rd ⊃ Ω be nonempty, bounded open set and let K ⊂ Rl be compact.
Let (f ) be a sequence of functions from Ω to K, convergent to some function
n
f weakly∗ inL∞. Finally,letϕbeanarbitrarycontinuousrealvaluedfunction
0
on Rd. Then the sequence (ϕ(f )) is uniformly bounded in L∞ norm and
n
thereforebytheBanach–Alaoglutheoremthereexistsasubsequenceof(ϕ(f ))
n
weakly∗ convergent to some function g. In general g 6= ϕ((f )). L. C. Young
0
provedin [11], that there exists a subsequence of ϕ((f )), not relabelled, and a
n
2
family (ν ) of probability measures with supports suppν ⊆ K, such that
x x∈Ω x
∀ϕ∈C(Rd)∀w ∈L1(Ω) there holds
lim ϕ(f (x))w(x)dx = ϕ(s)ν (ds)w(x)dx := ϕ(x)w(x)dx.
n x
n→∞
Z Z Z Z
Ω Ω K Ω
This family of probability measures is today called a Young measure associated
with the sequence (f ).
n
In 1989 J. M. Ball proved the following theorem. Let Ω be a measurable
subset of Rd, v: [0,+∞)→[0,+∞) a continuous, nondecreasing function such
that lim v(t)=+∞. By ψ we denote a function ψ: Ω×Rl ∋(x,λ)→ψ(x,λ)∈R
t→∞
satisfying Carath´eodory conditions: it is measurable with respect to the first,
andcontinuouswithrespecttothesecondvariable. Considerfurtherasequence
(f ) of functions on Ω with values in Rl, satisfying the condition
n
sup v(|f (x)|)dx<+∞.
n
n
Z
Ω
Theorem 2.1 ([3]) Under the above assumptions, there exists a subsequence
of (f ), not relabelled, and a family (ν ) of probability measures, dependent
n x x∈Ω
measurably on x, such that if for any Carath´eodory function ψ the sequence
(ψ(x,f (x)) is weakly convergent in L1(Ω), then its weak limit is a function
n
ψ(x)= ψ(x,λ)dν (λ).
x
RZl
We now turn our attention to the presentation of the Young measures as in [10].
In general, Young measures can be looked at as the element of the space con-
jugate to the space L1(Ω,C(K)) of Bochner integrable functions on Ω ⊂ Rd
with values in C(K). The space L1(Ω,C(K)) is isometrically isomorphic to
thespaceCar(Ω,K;R)oftheCarath´eodoryfunctions,equippedwiththenorm
khk := sup|h(x,k)|dx.
Car
Ω k∈K
Let h ∈RL1(Ω,C(K)). Denote by U the set of all measurable functions on
Ω with values in K. Consider a mapping
i: U →L1(Ω,C(K))∗
defined by the formula
hi(f),hi:= h(x,f(x))dx.
Z
Ω
By Y(Ω,K) we will denote the weak∗ closure of the set i(U) in L1(Ω,C(K))∗:
Y(Ω,K):= L1(Ω,C(K))∗ ∋η :∃(f )⊂U :i(f )−−w−∗−→η .
n n
n→∞
n o
Denote by
3
• rca(K) – the space of regular, countably additive signed measures on K,
equipped with the norm kmk := |m|(Ω), where |·| stands in this
rca(K)
case for the total variation of the measure m. With this norm rca(K) is
a Banach space;
• rca1(K)– the subsetofrca(K) with elements being probabilitymeasures
on K;
• L∞(Ω,rca(K)) – the set of the weakly measurable mappings
w
ν: Ω ∋x→ν(x)∈rca(K).
We equip this set with the norm
kνkL∞w(Ω,rca(K)) :=esssup kν(x)krca(K) :x∈Ω .
By the Dunford – Pettis theorem thi(cid:8)s space is isometrical(cid:9)ly isomorphic
with the space L1(Ω,C(K))∗.
Now define an element η of L1(Ω,C(K))∗ by the formula
η: L1(Ω,C(K))∋h→hη,hi:= h(x,k)dν (k) dx,
x
ΩZ (cid:16)KZ (cid:17)
which in turn will be the value of the mapping
ψ: L∞(Ω,rca(K))∋ν →ψ(ν):=η ∈L1(Ω,C(K))∗
w
Theorem 2.2 The mapping ψ defined above is an isometric isomorphism bet-
ween the spaces L∞(Ω,rca(K)) and L1(Ω,C(K))∗.
w
The set of the Young measures on the compact set K ⊂ Rl will be denoted by
Y(Ω,K):
Y(Ω,K):= ν =(ν(x))∈L∞(Ω,rca(K)):ν ∈rca1(K)for a.a x∈Ω .
w x
We will write νx(cid:8)or (νx)x∈Ω instead of ν(x). (cid:9)
Finally, we define the Dirac mapping δ: ∀x∈Ω
δ: U ∋f →[δ(f)](x):=δ ∈Y(Ω,K)
f(x)
Theorem 2.3 The diagram
||②②②②②δ②②②②②②②②②②UU ❊❊❊❊❊❊❊❊❊i❊❊❊❊❊❊""
oo //
YY((ΩΩ;;KK)) YY((ΩΩ;;KK))
ψ
is commutative.
This means, that for any f ∈U there exists a Young measure associated with it.
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2.2 An auxiliary lemma
The notion of quasi-Young measure was introduced in [9]. We state it now in a
slightly more general form.
Let Ω be an open subset of Rd with Lebesgue measure M >0, dµ(x):= 1 dx,
M
wheredxisthed–dimensionalLebesguemeasureonΩ andletK ⊂Rl becom-
pact.
Definition 2.1 We say that a family of probability measures ν = (ν ) is
x x∈Ω
a quasi-Young measure associated with the measurable function f: Rd ⊃ Ω →
K ⊂Rl, if for every continuous function β: K →R there holds an equality
β(k)dν (k)= β(f(x))dµ(x).
x
Z Z
K Ω
In [9] it was proved that in many application important cases the quasi-
Young measures associated with functions satysfying appropriate assumptions
were identical with the Young measures with them. In particular, quasi-Young
measures associated with simple functions are Young measures associated with
them. Denote by µ the normalized Lebesgue measure on a nonempty, bounded
subset Ω of Rd with positive measure M: dµ(x) := 1 dx with a d-dimensional
M
Lebesguemeasuredx. Let{Ω}n beapartitionofΩintoopen,pairwisedisjoint
i=1
n
subsets Ω with Lebesgue measure m >0, such that cl(Ω )=cl(Ω), where
i i i
i=1
’cl’ stands for ’closure’. By 1A we denote the characteSristic function of the set A.
Theorem 2.4 [9] Choose and fix points p ∈Rl, i=1,2,...,n, and let f be a
i
simple function:
n
f := p 1 .
i Ωi
i=1
X
Then the Young measure associated with f is of the form
n
1
ν = m δ .
x M i pi
i=1
X
Remark 2.1 Observe that ν does not depend on the variable x∈Ω; this is
x
a homogeneous Young measure.
Definition 2.2 The(quasi-)Youngmeasureassociatedwithsimplefunctionwill
be called a simple (quasi-)Young measure.
We now recall the notion of weak∗ convergence of measures on compact sets.
Definition 2.3 Wesay that asequence(ν )of boundedmeasures on acompact
n
set K ⊂Rl converges weakly∗ to a measure ν , if ∀β ∈C(K,R) there holds
0
lim β(k)dν (k)= β(k)dν (k).
n 0
n→∞
Z Z
K K
5
We now prove a lemma which needed further.
Lemma 2.1 Letf: Ω →K beameasurablefunctionandlet(f )beapointwise
n
convergent to f sequence of simple functions. Then the Young measure νf
associated with f is a weak∗ limit of the sequence of the simple Young measures
associated with respective elements of (f ).
n
Proof. Choose and fix ε > 0. Using change of variable theorem, continuity of
the function β and the finiteness of the measure of Ω, we infer the existence of
n ∈N such that ∀m, n>n we have
0 0
β(k)dν − β(k)dν = β(f (x))dµ− β(f (x))dµ ≤
n m n m
(cid:12)KZ KZ (cid:12) (cid:12)ΩZ ΩZ (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
≤ |β(f (x))−β(f (x))|dµ ≤ε·µ(Ω).
n m
Z
Ω
This means that (ν ) is a weak∗ Cauchy sequence, so its weak∗ accumulation
n
point belongs to Y(Ω,K). By the injectivity of ψ, there is exactly one Young
measure ρ = (ρ ) ∈ Y(Ω,K) corresponding to this accumulation point.
x x∈Ω
Then we have
β(k)dρ − β(k)dνf ≤ β(k)dρ − β(k)dν +
x x x n
(cid:12)KZ KZ (cid:12) (cid:12)KZ KZ (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
+ β(k)dν − β(k)dνf .
n x
(cid:12)KZ KZ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Since ρ is an accumulationpoint of (ν ) and (f ) convergespointwise to f, the
n n
result follows. (cid:4)
Remark 2.2 Observe that ν need not be a homogeneous Young measure.
f
Corollary 2.1 The set of all simple Young measures is weak∗ dense in the set
of the Young measures associated with functions from U.
3 Some necessary notions from probability the-
ory and notation
To set up notation, we recall now standard probabilistic notions needed in the
sequel. If Σ is a σ-algebra of subsets of a nonempty set A and P – a measure
on Σ, then the triple (A, Σ, P) is called a measure space, and a probability
space if P is a probability measure. A random variable (or a random vector)
X: A → Rd is a function such that for any Borel set B ⊆ Rd there holds
X−1(B) ∈ Σ. Obviously, if ϕ: Rd → Rl is a Borel function, then ϕ(X) is a
random variable. The probability distribution on Rd is any probability measure
6
P ontheσ-algebraB(Rd)ofBorelsubsetsofRd. Theprobabilitydistributionof
arandomvariableX withvaluesinRdisaprobabilitymeasureP onRddefined
X
for any B ∈ B(Rd) by the equality P (B) := P(X−1(B)). Consequently, for
X
the distribution of the random variable ϕ(X) we have: for any C ∈B(Rl)
P (C)=P(ϕ(X)−1(C))=P (ϕ−1(C)).
ϕ(X) X
If P is a probability distribution on Rd and for some Lebesgue integrable func-
tion g: Rd → R there holds: ∀A ∈ B(Rd) P(A) = g(x)dx, then the function
A
g is called a density of P. R
Let Ω be a Borel subset of Rd with Lebesgue measure M >0. We say that
random variable U: Rd →Rl is uniform on Ω, if its density g is of the form
u
1 , x∈Ω
g (x)= M
U
(0, x∈/ Ω.
The probability distribution P is then called the uniform distribution.
U
4 Main result
Asintheprevioussections,letΩbeanopensubsetofRdwithLebesguemeasure
M > 0, dµ(x) := 1 dx, where dx is the d – dimensional Lebesgue measure on
M
Ω and let K ⊂Rl be compact. Denote P := 1 dx.
M
Finally, we are ready to formulate the main theorem of the article.
Theorem 4.1 Let f: Rd ⊃ Ω → K ⊂ Rl be a Borel function with Young
measure µf. Then µf is the probability distribution of the random variable
Y =f(U), where U has a uniform distribution on Ω.
Proof. The distribution of a random variable Y is of the form: ∀C ∈ B(K),
P (C) = P (f−1(C)). Let f be constant on Ω with value p and vanish on
f(U) U
the complement of Ω. By lemma 2.1 we haveµf =δ . For any C ⊆K we have
p
1, p∈C
µf(C)= 1 (p)dδ =
C p
(0, p∈/ C.
RZl
On the other hand,
1 1 ·M =1, p∈C
P (f−1(C))= g dP = dx= M
U f−1Z(C) u M{x:f(Zx)∈C} (M1 ·0=0, p∈/ C.
ThusP =µf. This equalityalsoholds whenf is asimple function, due to the
Y
linearity of the integral. Since functions under consideration have values in the
compact set K, the dominated convergence theorem yields the result for any
Borel f. (cid:4)
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