## Brouwer Fixed Point Theorem in L ^{0} ^{d}

### Samuel Drapeau

^{a,1,∗}

### , Martin Karliczek

^{a,2,†}

### , Michael Kupper

^{b,3,∗}

### , Martin Streckfuß

^{a,4,‡}

### September 12, 2013

ABSTRACT

The classical Brouwer fixed point theorem states that inR^{d}
every continuous function from a convex, compact set on
itself has a fixed point. For an arbitrary probability space,
letL^{0}=L^{0}(Ω,A, P)be the set of random variables. We
consider(L^{0})^{d}as anL^{0}-module and show that local, se-
quentially continuous functions onL^{0}-convex, closed and
bounded subsets have a fixed point which is measurable by
construction.

KEYWORDS: Conditional Simplex, Fixed Points in(L^{0})^{d},
Brouwer

AUTHORSINFO

aHumboldt-Universität Berlin, Unter den Linden 6, 10099 Berlin, Ger- many

bUniversität Konstanz, Universitätsstraße 10, 78464 Konstanz, Germany

1drapeau@math.hu-berlin.de

2karliczm@math.hu-berlin.de

3kupper@uni-konstanz.de

4streckfu@math.hu-berlin.de

∗Funding: MATHEON project E.11

†Funding: Konsul Karl und Dr. Gabriele Sandmann Stiftung

‡Funding: Evangelisches Studienwerk Villigst PAPERINFO ArXiv ePrint:1305.2890

AMS CLASSIFICATION:47H10, 13C13, 46A19, 60H25 We thank Asgar Jamneshan for fruitful discussions.

### Introduction

The Brouwer fixed point theorem states that a continuous function from a compact and convex set inR^{d}
to itself has a fixed point. This result and its extensions play a central role in Analysis, Optimization and
Economic Theory among others. To show the result one approach is to consider functions on simplexes
first and use Sperner’s lemma.

Recently, Cheridito, Kupper, and Vogelpoth [4], inspired by the theory developed by Filipovi´c, Kup-
per, and Vogelpoth [7] and Guo [8], studied (L^{0})^{d} as an L^{0}-module, discussing concepts like linear
independence,σ-stability, locality andL^{0}-convexity. Based on this, we define affine independence and
conditional simplexes in(L^{0})^{d}. Showing first a result similar to Sperner’s Lemma, we obtain a fixed point
for local, sequentially continuous functions on conditional simplexes. From the measurable structure of
the problem, it turns out that we have to work with local, measurable labeling functions. To cope with this
difficulty and to maintain some uniform properties, we subdivide the conditional simplex barycentrically.

We then prove the existence of a measurable completely labeled conditional simplex, contained in the original one, which turns out to be a suitableσ-combination of elements of the barycentric subdivision along a partition ofΩ. Thus, we can construct a sequence of conditional simplexes converging to a point.

By applying always the same rule of labeling using the locality of the function, we show that this point is
a fixed point. Due to the measurability of the labeling function the fixed point is measurable by construc-
tion. Hence, even though we follow the approach ofR^{d}(cf. [2]) we do not need any measurable selection
argument.

In Probabilistic Analysis theory the problem of finding random fixed points of random operators is an important issue. GivenC, a compact convex set of a Banach space, a continuous random operator is a functionR: Ω× C → Csatisfying

(i) R(., x) : Ω→ Cis a random variable for any fixedx∈ C, (ii) R(ω, .) :C → Cis a continuous function for any fixedω∈Ω.

For R there exists a random fixed point which is a random variable ξ: Ω → C such that ξ(ω) =
R(ω, ξ(ω))for anyω(cf. [1], [9], [6]). In contrast to thisω-wise consideration, our approach is com-
pletely within the theory ofL^{0}. All objects and properties are therefore defined in that language and
proofs are done withL^{0}-methods. Moreover, the connection between continuous random operators on
R^{d}and sequentially continuous functions on(L^{0})^{d}is not entirely clear.

An application, though not studied in this paper, is for instance possible in economic theory or op-
timization in the context of [3]. Therein methods from convex analysis are used to obtain equilibrium
results for translation invariant utility functionals on(L^{0})^{d}. Without translation invariance these methods
fail, and will be replaced by fixed point arguments in an ongoing work. Thus, our result is helpful to
develop the theory of non-translation invariant preference functionals mapping toL^{0}.

The present paper is organized as follows. In the first chapter we present the basic concepts concerning
(L^{0})^{d} as anL^{0}-module. We define conditional simplexes and examine their basic properties. In the
second chapter we define measurable labeling functions and show the Brouwer fixed point theorem for
conditional simplexes via a construction in the spirit of Sperner’s lemma. In the third chapter, we show a
fixed point result forL^{0}-convex, bounded and sequentially closed sets in(L^{0})^{d}. With this result at hand,
we present the topological implications known from the real-valued case. On the one hand, we show
the impossibility of contracting a ball to a sphere in(L^{0})^{d}and on the other hand, an intermediate value
theorem inL^{0}.

### 1 Conditional Simplex

For a probability space(Ω,A, P), let L^{0} = L^{0}(Ω,A, P)be the space of all A-measurable random
variables, whereP-almost surely equal random variables are identified. In particular, forX, Y ∈ L^{0},
the relationsX ≥Y andX > Y have to be understoodP-almost surely. The setL^{0}with theP-almost
everywhere order is a lattice ordered ring, and every nonempty subsetC ⊆ L^{0} has a least upper bound
ess supC and a greatest lower boundess infC. Form ∈ R, we denote the constant random variable
m1_{Ω}bym. Further, we define the setsL^{0}_{+} = {X ∈ L^{0}:X ≥ 0},L^{0}_{++} = {X ∈ L^{0}:X > 0}and
A+ ={A ∈ A:P(A) >0}. The set of random variables with values in a setM ⊆Ris denoted by
M(A). For example,{1, . . . , r}(A)is the set ofA-measurable functions with values in{1, . . . , r} ⊆N,
[0,1](A) ={Z∈L^{0}: 0≤Z ≤1}and(0,1)(A) ={Z∈L^{0}: 0< Z <1}.

Theconvex hullofX_{1}, . . . , X_{N} ∈(L^{0})^{d},N∈N, is defined as

conv (X1, . . . , XN) =
( _{N}

X

i=1

λiXi:λi∈L^{0}_{+},

N

X

i=1

λi= 1 )

.

An elementY = PN

i=1λ_{i}X_{i} such that λ_{i} > 0 for alli ∈ I ⊆ {1, . . . , N}is called astrict convex
combinationof{Xi:i∈I}.

Theσ-stable hullof a setC ⊆(L^{0})^{d}is defined as

σ(C) = (

X

i∈N

1_{A}_{i}X_{i}:X_{i}∈ C,(A_{i})_{i∈}_{N}is apartition
)

,

where a partition is a countable family (Ai)_{i∈}_{N} ⊆ A such that P(Ai ∩Aj) = 0 for i 6= j and
P(S

i∈NAi) = 1. We call a nonempty setCσ-stableif it is equal toσ(C). For aσ-stable setC ⊆(L^{0})^{d}a
functionf:C →(L^{0})^{d}is calledlocaliff(P

i∈N1A_{i}Xi) =P

i∈N1A_{i}f(Xi)for every partition(Ai)i∈N

andXi ∈ C,i ∈ N. ForX,Y ⊆ (L^{0})^{d}, we call a function f: X → Ysequentially continuousif for
every sequence(X_{n})_{n∈}_{N}inX converging toX ∈ X P-almost-surely it holds thatf(X_{n})converges to

f(X)P-almost surely. Further, theL^{0}-scalar productandL^{0}-normon(L^{0})^{d}are defined as

hX, Yi=

d

X

i=1

X_{i}Y_{i} and kXk=hX, Xi^{1}^{2}.

We call C ⊆ (L^{0})^{d} bounded if ess sup_{X∈C}kXk ∈ L^{0} andsequentially closed if it contains allP-
almost sure limits of sequences inC. Further, the diameter of C ⊆ (L^{0})^{d} is defined asdiam(C) =
ess sup_{X,Y}_{∈C}kX−Yk.

Definition 1.1. ElementsX_{1}, . . . , X_{N} of (L^{0})^{d}, N ∈ N, are said to be affinely independent, if either
N = 1orN >1and{X_{i}−X_{N}}^{N}_{i=1}^{−1}arelinearly independent, that is

N−1

X

i=1

λi(Xi−XN) = 0 implies λ1=· · ·=λN−1= 0, (1.1)

whereλ1, . . . , λ_{N−1}∈L^{0}.

The definition of affine independence is equivalent to

N

X

i=1

λiXi= 0 and

N

X

i=1

λi= 0 implies λ1=· · ·=λN = 0. (1.2)

Indeed, first, we show that (1.1) implies (1.2). Let PN

i=1λiXi = 0 and PN

i=1λi = 0. Then, PN−1

i=1 λ_{i}(X_{i}−X_{N}) = λ_{N}X_{N} +PN−1

i=1 λ_{i}X_{i} = 0. By assumption (1.1),λ_{1} = · · · = λ_{N−1} = 0,
thus alsoλN = 0. To see that (1.2) implies (1.1), letPN−1

i=1 λi(Xi−XN) = 0. WithλN =−PN−1 i=1 λi, it holds PN

i=1λiXi = λNXN +PN−1

i=1 λiXi = PN−1

i=1 λi(Xi −XN) = 0. By assumption (1.2), λ1=· · ·=λN = 0.

Remark 1.2. We observe that if(X_{i})^{N}_{i=1}⊆(L^{0})^{d}are affinely independent then(λX_{i})^{N}_{i=1}, forλ∈L^{0}_{++},
and (X_{i} +Y)^{N}_{i=1}, for Y ∈ (L^{0})^{d}, are affinely independent. Moreover, if a family X_{1}, . . . , X_{N} is
affinely independent then also1BX1, . . . ,1BXN are affinely independent onB ∈ A+, which means
fromPN

i=11BλiXi= 0andPN

i=11Bλi= 0always follows1Bλi= 0for alli= 1, . . . , N.
Definition 1.3. Aconditional simplexin(L^{0})^{d}is a set of the form

S= conv(X_{1}, . . . , X_{N})

such thatX_{1}, . . . , X_{N} ∈(L^{0})^{d}are affinely independent. We callN∈Nthe dimension ofS.

Remark 1.4. In a conditional simplexS = conv(X1, . . . , XN), the coefficients of convex combinations are unique in the sense that

N

X

i=1

λiXi=

N

X

i=1

µiXiand

N

X

i=1

λi =

N

X

i=1

µi = 1 implies λi =µifor alli= 1, . . . , N. (1.3)

Indeed, sincePN

i=1(λi−µi)Xi = 0andPN

i=1(λi−µi) = 0, it follows from (1.2) thatλi−µi= 0for

alli= 1, . . . , N.

Remark 1.5. Note that the present setting —L^{0}-modules and the sequentialP-almost sure convergence

— is of local nature. This is for instance, not the case for subsets ofL^{p} or the convergence in theL^{p}-
norm for1≤p <∞. First,L^{p}is not closed under multiplication and hence neither a ring nor a module
over itself so that we can not even speak about affine independence. Second, it is mostly not aσ-stable
subspace ofL^{0}. However, for a conditional simplexS= conv(X_{1}, . . . , X_{N})in(L^{0})^{d}such that anyX_{k}is
in(L^{p})^{d}, it holds thatSis uniformly bounded byNsup_{k=1,...,N}kXkk ∈L^{p}. This uniform boundedness
yields that anyP-almost sure converging sequence inSis also converging in theL^{p}-norm for1≤p <∞
due to the dominated convergence theorem. This shows how one can translate results fromL^{0}toL^{p}.
Since a conditional simplex is a convex hull it is in particularσ-stable. In contrast to a simplex inR^{d}
the representation ofSas a convex hull of affinely independent elements is unique but up toσ-stability.

Proposition 1.6. Let(Xi)^{N}_{i=1}and(Yi)^{N}_{i=1}be families in(L^{0})^{d}withσ(X1, . . . , XN) =σ(Y1, . . . , YN).

Thenconv(X1, . . . , XN) = conv(Y1, . . . , YN). Moreover,(Xi)^{N}_{i=1}are affinely independent if and only
if(Yi)^{N}_{i=1}are affinely independent.

IfS is a conditional simplex such thatS = conv(X1, . . . , XN) = conv(Y1, . . . , YN), then it holds
σ(X_{1}, . . . , X_{N}) =σ(Y_{1}, . . . , Y_{N}).

Proof. Supposeσ(X1, . . . , XN) =σ(Y1, . . . , YN). Fori= 1, . . . , N, it holds Xi∈σ(X1, . . . , XN) =σ(Y1, . . . , YN)⊆conv(Y1, . . . , YN).

Therefore,conv(X1, . . . , XN)⊆conv(Y1, . . . , YN)and the reverse inclusion holds analogously.

Now, let(X_{i})^{N}_{i=1} be affinely independent andσ(X_{1}, . . . , X_{N}) = σ(Y_{1}, . . . , Y_{N}). We want to show
that(Y_{i})^{N}_{i=1}are affinely independent. To that end, we define the affine hull

aff(X1, . . . , XN) =
( _{N}

X

i=1

λiXi:λi∈L^{0},

N

X

i=1

λi = 1 )

.

First, letZ_{1}, . . . , Z_{M} ∈ (L^{0})^{d}, M ∈ N, such that σ(X_{1}, . . . , X_{N}) = σ(Z_{1}, . . . , Z_{M}). We show
that if1_{A}aff(X_{1}, . . . , X_{N})⊆1_{A}aff(Z_{1}, . . . , Z_{M})forA∈ A_{+}andX_{1}, . . . , X_{N} are affinely indepen-
dent then M ≥ N. SinceXi ∈ σ(X1, . . . , XN) = σ(Z1, . . . , ZM) ⊆ aff(Z1, . . . , ZM), we have
aff(X1, . . . , XN)⊆aff(Z1, . . . , ZM). Further, it holds thatX1=PM

i=11_{B}1

iZifor a partition(B^{1}_{i})^{M}_{i=1}
and hence there exists at least oneB_{k}^{1}

1 such that A^{1}_{k}

1 := B^{1}_{k}

1 ∩A ∈ A_{+}, and1_{A}1
k1

X_{1} = 1_{A}1
k1

Z_{k}_{1}.
Therefore,

1_{A}1
k1

aff(X1, . . . , XN)⊆1_{A}1
k1

aff(Z1, . . . , ZM) = 1_{A}1
k1

aff({X1, Z1, . . . , ZM} \ {Zk_{1}}).

ForX2 =PM
i=11_{A}2

iZiwe find a setA^{2}_{k}, such thatA^{2}_{k}

2 =A^{2}_{k}∩A^{1}_{k}

1 ∈ A+,1_{A}2
k2

X2 = 1_{A}2
k2

Zk_{2} and
k1 6= k2. Assume to the contraryk2 = k1, then there exists a setB ∈ A+, such that1BX1 = 1BX2

which is a contradiction to the affine independence of(Xi)^{N}_{i=1}. Hence, we can again substituteZk_{2} by
X2onA^{2}_{k}_{2}. Inductively, we findk1, . . . , kN such that

1A_{kN}aff(X1, . . . , XN)⊆1A_{kN} aff({X1, . . . , XN, Z1, . . . , ZM} \ {Zk_{1}, . . . Zk_{N}})

which showsM ≥ N. Now supposeY1, . . . , YN are not affinely independent. This means, there ex-
ist(λi)^{N}_{i=1} such that PN

i=1λiYi = PN

i=1λi = 0 but not all coefficientsλi are zero, without loss of
generality,λ_{1} > 0 onA ∈ A_{+}. Thus,1_{A}Y_{1} = −1_{A}PN

i=2
λ_{i}

λ_{1}Y_{i} and it holds1_{A}aff(Y_{1}, . . . , Y_{N}) =

1Aaff(Y2, . . . , YN). To see this, consider 1AZ = 1APN

i=1µiYi ∈ 1Aaff(Y1, . . . , YN)which means
1_{A}PN

i=1µ_{i}= 1_{A}. Thus, inserting forY_{1},

1AZ= 1A

"_{N}
X

i=2

µiYi−µ1 N

X

i=2

λi

λ1

Yi

#

= 1A

"_{N}
X

i=2

µi−µ1

λi

λ1

Yi

# .

Moreover,

1A

" _{N}
X

i=2

µi−µ1

λi

λ_{1}
#

= 1A

" _{N}
X

i=2

µi

# + 1A

"

−µ1

λ_{1}

N

X

i=2

λi

#

= 1A(1−µ1) + 1A

µ1

λ_{1}λ1= 1A.
Hence, 1AZ ∈ 1Aaff(Y2, . . . , YN). It follows that 1Aaff(X1, . . . , XN) = 1Aaff(Y1, . . . , YN) =
1Aaff(Y2, . . . , YN). This is a contradiction to the former part of the proof (becauseN−16≥N).

Next, we characterize extremal points inS = conv(X_{1}, . . . , X_{N}). To this end, we show X ∈
σ(X_{1}, . . . , X_{N})if and only if there do not exist Y and Z in S \ {X} andλ ∈ (0,1)(A) such that
λY + (1−λ)Z = X. ConsiderX ∈ σ(X1, . . . , XN)which isX = PN

k=11A_{k}Xk for a partition
(Ak)k∈N. Now assume to the contrary that we findY =PN

k=1λkXkandZ=PN

k=1µkXkinS \ {X} such thatX=λY + (1−λ)Z. This means thatX =PN

k=1(λλ_{k}+ (1−λ)µ_{k})X_{k}. Due to uniqueness of
the coefficients (cf. (1.3)) in a conditional simplex we haveλλ_{k}+ (1−λ)µ_{k} = 1_{A}_{k}for allk= 1. . . , N.

By means of0 < λ < 1, it holds thatλλk + (1−λ)µk = 1A_{k} if and onlyλk = µk = 1A_{k}. Since
the last equality holds for allk it follows thatY = Z = X. Therefore, we cannot find Y andZ in
S \ {X}such thatX is a strict convex combination of them. On the other hand, considerX ∈ S such
thatX /∈σ(X_{1}, . . . , X_{N}). This means,X =PN

k=1ν_{k}X_{k}, such that there existν_{k}_{1}andν_{k}_{2}andB∈ A+

with0 < ν_{k}_{1} < 1on B and0 < ν_{k}_{2} < 1 onB. Define ε := ess inf{ν_{k}_{1}, ν_{k}_{2},1−ν_{k}_{1},1−ν_{k}_{2}}.

Then defineµk =λk = νk ifk1 6=k 6=k2andλk_{1} = νk_{1} −ε,λk_{2} = νk_{2} +ε,µk_{1} = νk_{1} +εand
µk_{2} =νk_{2}−ε. Thus,Y =PN

k=1λkXk andZ =PN

k=1µkXkfulfill0.5Y + 0.5Z =X but both are not equal toX by construction. Hence,Xcan be written as a strict convex combination of elements in S \ {X}. To conclude, considerX∈σ(X1, . . . , XN)⊆ S= conv(X1, . . . , XN) = conv(Y1, . . . , YN).

SinceX∈σ(X_{1}, . . . , X_{N})it is not a strict convex combinations of elements inS \ {X}, in particular, of
elements inconv(Y_{1}, . . . , Y_{N})\ {X}. Therefore,Xis also inσ(Y_{1}, . . . , Y_{N}). Hence,σ(X_{1}, . . . , X_{N})⊆
σ(Y1, . . . , YN). With the same argumentation the other inclusion follows.

As an example let us consider[0,1](A). For an arbitraryA∈ A, it holds that1Aand1A^{c}are affinely
independent andconv(1A,1A^{c}) ={λ1A+ (1−λ)1A^{c}: 0≤λ≤1}= [0,1](A). Thus, the conditional
simplex[0,1](A)can be written as a convex combination of different affinely independent elements of
L^{0}. This is due to the fact thatσ(0,1) ={1B :B∈ A}=σ(1_{A},1_{A}c)for allA∈ A.

Remark 1.7. In(L^{0})^{d}, lete_{i}be the random variable which is1in thei-th component and0in any other.

Then the family0, e1, . . . , edis affinely independent and(L^{0})^{d}= aff(0, e1, . . . , ed). Hence, the maximal

number of affinely independent elements in(L^{0})^{d}isd+ 1.

The characterization ofX ∈σ(X1, . . . , XN)leads to the following definition.

Definition 1.8. LetS = conv(X_{1}, . . . , X_{N})be a conditional simplex. We define the set of extremal
pointsext(S) = σ(X_{1}, . . . , X_{N}). For an index set I and a collection S = (S_{i})_{i∈I} of conditional
simplexes we denoteext(S) =σ(ext(Si) :i∈I).

Remark 1.9. LetS^{j} = conv(X_{1}^{j}, . . . , X_{N}^{j}),j ∈ N, be conditional simplexes of the same dimension
N and(Aj)j∈Na partition. ThenP

j∈N1A_{j}S^{j} is again a conditional simplex. To that end, we define

Yk=P

j∈N1A_{j}X_{k}^{j}and recognizeP

j∈N1A_{j}S^{j}= conv(Y1, . . . , YN). Indeed,

N

X

k=1

λ_{k}Y_{k}=

N

X

k=1

λ_{k}X

j∈N

1_{A}_{j}X_{k}^{j} =X

j∈N

1_{A}_{j}

N

X

k=1

λ_{k}X_{k}^{j}∈X

j∈N

1_{A}_{j}S^{j}, (1.4)

showsconv(Y1, . . . , YN) ⊆P

j∈N1A_{j}S^{j}. The other inclusion follows by consideringPN

k=1λ^{j}_{k}X_{k}^{j} ∈
S^{j} and definingλk = P

j∈N1A_{j}λ^{j}_{k}. To show thatY1, . . . , YN are affinely independent, we consider
PN

k=1λkYk = 0 =PN

k=1λk. Then by (1.4), it holds1A_{j}PN

k=1λkX_{k}^{j} = 0and sinceS^{j}is a conditional
simplex,1Ajλk = 0for allj∈Nandk= 1, . . . , N. From the fact that(Aj)j∈Nis a partition, it follows

thatλ_{k}= 0for allk= 1, . . . , N.

We will prove the Brouwer fixed point theorem in the present setting using an analogue version of Sperner’s lemma. As in the unconditional case we have to subdivide a conditional simplex in smaller ones. For our argumentation we cannot use arbitrary subdivisions and need very special properties of the conditional simplexes in which we subdivide. This leads to the following definition.

Definition 1.10. LetS = conv(X_{1}, . . . , X_{N})be a conditional simplex andS_{N}the group of permutations
of{1, . . . , N}. Then forπ∈SNwe define

Cπ= conv

X_{π(1)},X_{π(1)}+X_{π(2)}

2 , . . . ,X_{π(1)}+· · ·+X_{π(k)}

k , . . . ,X_{π(1)}+· · ·+X_{π(N}_{)}
N

.

We call(Cπ)π∈S_{N} thebarycentric subdivisionofS, and denoteY_{k}^{π}= ^{1}_{k}Pk

i=1Xπ(i).

Lemma 1.11. The barycentric subdivision is a collection of finitely many conditional simplexes satisfying the following properties

(i) σ(S

π∈SNCπ) =S.

(ii) Cπhas dimensionN,π∈S_{N}.

(iii) Cπ∩ Cπis a conditional simplex of dimensionr∈Nandr < N forπ, π∈SN,π6=π.

(iv) Fors= 1, . . . , N−1, letBs:= conv(X1, . . . , Xs). All conditional simplexesCπ∩ Bs,π∈SN, of dimensionssubdivideBsbarycentrically.

Proof. We show the affine independence ofY_{1}^{π}, . . . , Y_{N}^{π}inCπ. It holds

λ_{π(1)}X_{π(1)}+λ_{π(2)}X_{π(1)}+X_{π(2)}

2 +· · ·+λ_{π(N)}
PN

k=1X_{π(k)}

N =

N

X

i=1

µ_{i}X_{i},
withµi = PN

k=π^{−1}(i)
λ_{π(k)}

k . Since PN

i=1µi = PN

i=1λi, the affine independence ofY_{1}^{π}, . . . , Y_{N}^{π} is
obtained by the affine independence ofX1, . . . , XN. Therefore allCπare conditional simplexes.

The intersection of two conditional simplexesCπandCπcan be expressed in the following manner. Let J ={j:{π(1), . . . , π(j)}={π(1), . . . , π(j)}}be the set of indexes up to which bothπandπhave the same set of images. Then,

C_{π}∩ C_{π}= conv
Pj

k=1X_{π(k)}
j :j∈J

!

. (1.5)

To show ⊇, let j ∈ J. It holds that

Pj
k=1X_{π(k)}

j is in both Cπ andCπ since{π(1), . . . , π(j)} = {π(1), . . . , π(j)}. Since the intersection of convex sets is convex, we get this implication.

As for the reverse inclusion, consider X ∈ Cπ∩ Cπ. FromX ∈ Cπ ∩C¯π, it follows thatX = PN

i=1λi(Pi k=1

X_{π(k)}

i ) =PN

i=1µi(Pi k=1

X_{π(k)}

i ). Considerj6∈J. By definition ofJ, there existp, q≤
jwithπ^{−1}(π(p)), π^{−1}(π(q)) 6∈ {1, . . . , j}. By (1.3), the coefficients ofX_{π(p)}are equal: PN

i=p
λ_{i}

i = PN

i=π^{−1}(π(p))
µ_{i}

i. The same holds forXπ(q):PN i=q

µ_{i}
i =PN

i=π^{−1}(π(q))
λi

i . Put together

N

X

i=j+1

µi

i ≤

N

X

i=q

µi

i =

N

X

i=π^{−1}(π(q))

λi

i ≤

N

X

i=j+1

λi

i ≤

N

X

i=p

λi

i =

N

X

i=π^{−1}(π(p))

µi

i ≤

N

X

i=j+1

µi

i
which is only possible ifµ_{j}=λ_{j}= 0sincep, q≤j.

Furthermore, ifCπ∩ Cπis of dimensionNby (1.5) follows thatπ=π. This shows(iii).

As for Condition(i), it clearly holdsσ(∪_{π∈S}_{N}Cπ)⊆ S. On the other hand, letX =PN

i=1λiXi ∈ S.

Then, we find a partition(An)_{n∈N}such that on everyAn the indexes are completely ordered which is
λi^{n}_{1} ≥λi^{n}_{2} ≥ · · · ≥λi^{n}_{N} onAn. This means, thatX ∈1A_{n}Cπ^{n}withπ^{n}(j) =i^{n}_{j}. Indeed, we can rewrite
XonA_{n}as

X = (λi^{n}_{1} −λi^{n}_{2})Xi^{n}_{1} +· · ·+ (N−1)(λi^{n}_{N−1}−λi^{n}_{N})
PN−1

k=1 X_{i}^{n}

k

N−1 +N λi^{n}_{N}

PN
k=1X_{i}^{n}

k

N ,

which shows thatX ∈ Cπ^{n}onA_{n}.

Further, forB_{s}= conv(X_{1}, . . . , X_{s})the elementsC_{π}^{0}∩ B_{s}of dimensionsare exactly the ones with
{π^{0}(i) :i= 1, . . . , s}={1, . . . , s}. To this end, letCπ^{0}∩ Bsbe of dimensions. This means there exists
an elementY in this intersection such thatY =PN

i=1λiXiwithλi>0for alli= 1, . . . , sandλi= 0
fori > s. As an element ofCπ^{0} thisY has a representation of the formY =PN

j=1(PN k=j

µ_{k}

k )Xπ^{0}(j),
forPN

k=1µ_{k}= 1andµ_{k} ∈L^{0}_{+}for everyk= 1, . . . , N. Suppose now that there exists somej_{0}≤swith
π^{0}(j0)> s. Then due toλ_{π}0(j0)= 0and the uniqueness of the coefficients (cf.(1.3)) in anL^{0}-simplex, it
holdsPN

k=j_{0}
µ_{k}

k = 0and withinPN k=j

µ_{k}

k = 0for allj ≥j0. This meansY =Pj_{0}−1
j=1 (PN

k=j
µ_{k}

k )Xπ^{0}(j)

and henceY is the convex combination ofj_{0}−1elements withj_{0}−1< s. This contradicts the property
thatλ_{i} >0forselements. Therefore,(C_{π}^{0} ∩ B_{s})_{π}^{0} is exactly the barycentric subdivision ofB_{s}, which

has been shown to fulfill the properties(i)-(iii).

Subdividing a conditional simplexS = conv(X_{1}, . . . , X_{N})barycentrically we obtain(C_{π})_{π∈S}_{N}. Di-
viding everyCπ barycentrically results in a new collection of conditional simplexes and we call this
the two-fold barycentric subdivision ofS. Inductively, we can subdivide every conditional simplex of
the(m−1)th step barycentrically and call the resulting collection of conditional simplexes them-fold
barycentric subdivision ofS and denote it byS^{m}. Further, we define ext(S^{m}) = σ({ext(C) : C ∈
S^{m}})to be theσ-stable hull of all extremal points of the conditional simplexes of them-fold barycen-
tric subdivision ofS. Notice that this is theσ-stable hull of only finitely many elements, since there are
only finitely many simplexes in the subdivision each of which is the convex hull ofNelements.

Remark 1.12. Consider an arbitraryCπ = conv(Y_{1}^{π}, . . . , Y_{N}^{π}),π∈SNin the barycentric subdivision of
a conditional simplexS. Then it holds

diam(Cπ)≤ ess sup

i,j=1,...,N

Y_{i}^{π}−Y_{j}^{π}

≤N−1

N diam(S).

Since this holds for anyπ ∈ SN, it follows that the diameter ofS^{m}, which is an arbitrary conditional
simplex of the m-fold barycentric subdivision ofS, fulfillsdiam(S^{m}) ≤ ^{N−1}_{N} ^{m}

diam(S). Since
diam(S)<∞and ^{N}_{N}^{−1}^{m}

→0, form→ ∞, it follows thatdiam(S^{m})→0, form→ ∞for every

sequence(S^{m})_{m∈}_{N}.

### 2 Brouwer Fixed Point Theorem for Conditional Simplexes

Definition 2.1. LetS= conv(X_{1}, . . . , X_{N})be a conditional simplex,m-fold barycentrically subdivided
inS^{m}. A local functionφ: ext(S^{m})→ {1, . . . , N}(A)is called alabeling functionofS. For fixed
X1, . . . , XN ∈ext(S)withS = conv(X1, . . . , XN), the labeling function is calledproper, if for any
Y ∈ext(S^{m})it holds that

P({φ(Y) =i} ⊆ {λi>0}) = 1, fori = 1, . . . , N, whereY = PN

i=1λiXi. A conditional simplexC = conv(Y1, . . . , YN) ⊆ S, with
Yj ∈ext(S^{m}), j= 1, . . . , N, is said to becompletely labeledbyφif this is a proper labeling function
ofSand

P

N

[

j=1

{φ(Yj) =i}

= 1,

for alli∈ {1, . . . , N}.

Lemma 2.2. LetS = conv(X_{1}, . . . , X_{N})be a conditional simplex andf:S → S be a local function.

Letφ: ext(S^{m})→ {0, . . . , N}(A)be a local function such that

(i) P({φ(X) =i} ⊆ {λi >0} ∩ {λi≥µi}) = 1, for alli= 1, . . . , N, (ii) P

SN

i=1 {λi >0} ∩ {λi≥µi}

⊆SN

i=1{φ(X) =i}

= 1, whereX=PN

i=1λiXiandf(X) =PN

i=1µiXi. Then,φis a proper labeling function.

Moreover, the set of functions fulfilling these properties is non-empty.

Proof. First we show thatφis a labeling function. Sinceφis local we just have to prove thatφactually maps to{1, . . . , N}. Due to(ii), we have to show thatP(SN

i=1{λi≥µ_{i}:λ_{i} >0}) = 1. Assume to the
contrary,µi > λi onA∈ A+, for allλiwithλi >0onA. Then it holds that1 = PN

i=1λi1_{{λ}_{i}_{>0}} <

PN

i=1µi1_{{µ}_{i}_{>0}} = 1onAwhich yields a contradiction. Thus,φis a labeling function. Moreover, due
to(i)it holds thatP({φ(X) =i} ⊆ {λi>0}) = 1which shows thatφis proper.

To prove the existence, for X ∈ ext(S^{m})withX = PN

i=1λ_{i}X_{i}, f(X) = PN

i=1µ_{i} let B_{i} :=

{λi > 0} ∩ {λi ≥ µi}, i = 1, . . . , N. Then we define the function φ at X as {φ(X) = i} = Bi \(Si−1

k=1Bk),i = 1, . . . , N. It has been shown thatφmaps to{1, . . . , N}(A) and is proper. It remains to show that φis local. To this end, considerX = P

j∈N1_{A}_{j}X^{j} whereX^{j} = PN
i=1λ^{j}_{i}X_{i}
and f(X^{j}) = PN

i=1µ^{j}_{i}X_{i}. Due to uniqueness of the coefficients in a conditional simplex it holds
that λi = P

j∈N1A_{j}λ^{j}_{i} and due to locality of f it follows that µi = P

j∈N1A_{j}µ^{j}_{i}. Therefore it
holds that B_{i} = S

j∈N

{λ^{j}_{i} >0} ∩ {λ^{j}_{i} ≥µ^{j}_{i}} ∩A_{j}

= S

j∈N(B^{j}_{i} ∩A_{j}). Hence, φ(X) = i on
Bi \ (Si−1

k=1Bk) = [S

j∈N(B^{j}_{i} ∩Aj)]\[Si−1
k=1(S

j∈NB_{k}^{j} ∩Aj)] = S

j∈N[(B_{i}^{j} \Si−1

k=1B_{k}^{j})∩ Aj].

On the other hand, we see that P

j∈N1_{A}_{j}φ(X^{j}) is i on any A_{j} ∩ {φ(X^{j}) = i}, hence it is i on
S

j∈N(B_{i}^{j}\Si−1

k=1B_{k}^{j})∩Aj. Thus,P

j∈N1A_{j}φ(X^{j}) =φ(P

j∈N1A_{j}X^{j})which shows thatφis local.

The reason to demand locality of a labeling function is exactly because we want to label by the rule
explained in Lemma2.2and hence keep local information with it. For example consider a conditional
simplexS= conv(X_{1}, X_{2}, X_{3}, X_{4})andΩ ={ω1, ω_{2}}. LetY ∈ext(S)be given byY =^{1}_{3}P3

i=1X_{i}.
Now consider a functionfonSsuch that

f(Y)(ω1) = 1

4X1(ω1) +3

4X3(ω1); f(Y)(ω2) =2

5X1(ω2) +2

5X2(ω2) +1

5X4(ω2).

If we labelY by the rule explained in Lemma2.2,φtakes the valuesφ(Y)(ω1)∈ {1,2}andφ(Y)(ω2) = 3. Therefore, we can really distinguish on which setsλi ≥µi. Yet, using a deterministic labeling ofY, we would loose this information.

Theorem 2.3. LetS= conv(X_{1}, . . . , X_{N})be a conditional simplex in(L^{0})^{d}. Letf:S → Sbe a local,
sequentially continuous function. Then there existsY ∈ Ssuch thatf(Y) =Y.

Proof. We consider the barycentric subdivision (Cπ)_{π∈S}_{N} of S and a proper labeling function φ on
ext(S). First, we show that we can find a completely labeled conditional simplex inS. By induc-
tion on the dimension ofS = conv(X1, . . . , XN), we show that there exists a partition(Ak)k=1,...,K

such that on anyA_{k} there is an odd number of completely labeledCπ. The caseN = 1is clear, since a
point can be labeled with the constant index1, only.

Suppose the caseN −1 is proven. Since the number ofY_{i}^{π} of the barycentric subdivision is finite
andφcan only take finitely many values, it holds for allV ∈(Y_{i}^{π})i=1,...,N,π∈SN there exists a partition
(A^{V}_{k})k=1,...,K,K <∞, whereφ(V)is constant on anyA^{V}_{k}. Therefore, we find a partition(Ak)k=1,...,K,
such thatφ(V)onAkis constant for allV andAk. FixAk now.

In the following, we denote byC_{π}b these conditional simplexes for whichC_{π}b∩ BN−1 are N −1-
dimensional (cf. Lemma1.11 (iv)), thereforeπ^{b}(N) =N. Further we denote byC_{π}c these conditional
simplexes which are not of the typeC_{π}b, that isπ^{c}(N)6=N. If we useCπwe mean a conditional simplex
of arbitrary type. We define

• C ⊆(Cπ)π∈S_{N} to be the set ofCπwhich are completely labeled onAk.

• A ⊆(Cπ)_{π∈S}_{N} to be the set of theP-almost completely labeledCπ, that is
{φ(Y_{k}^{π}), k= 1, . . . , N}={1, . . . , N −1} onA_{k}.

• Eπto be the set of the intersections(Cπ∩ Cπ_{l})π_{l}∈SNwhich areN−1-dimensional and completely
labeled onAk.^{1}

• Bπto be the set of the intersectionsCπ∩ BN−1which are completely labeled onAk.

It holds thatEπ∩Bπ =∅and hence|Eπ∪Bπ|=|Eπ|+|Bπ|. SinceCπ^{c}∩ BN−1is at mostN−2-
dimensional, it holds thatBπ^{c} =∅and hence|Bπ^{c}|= 0. Moreover, we know thatC_{π}∩ C_{π}_{l} isN−1-
dimensional onAk if and only if this holds on wholeΩ(cf. Lemma1.11 (iii)) andC_{π}b∩ BN−1 6=∅on
Ak if and only if this also holds on wholeΩ(cf. Lemma1.11 (iv)). So it does not play any role if we
look at these sets which are intersections onAkor onΩsince they are exactly the same sets.

IfCπ^{c} ∈C then|Eπ^{c}| = 1and ifC_{π}b ∈C then|Eπ^{b}∪Bπ^{b}|= 1. IfCπ^{c} ∈A then|Eπ^{c}|= 2and if
C_{π}b ∈A then|Eπ^{b}∪Bπ^{b}|= 2. Therefore it holdsP

π∈SN|Eπ∪Bπ|=|C|+ 2|A|.

If we pick anE_{π}∈Eπwe know there always exists exactly one otherπ_{l}such thatE_{π} ∈Eπl(Lemma
1.11(iii)). ThereforeP

π∈SN|Eπ|is even. Moreover(C_{π}b∩ BN−1)_{π}bsubdividesB_{N−1}barycentrically^{2}
and hence we can apply the hypothesis (onext(C_{π}b∩ B_{N−1})). This means that the number of completely
labeled conditional simplexes is odd on a partition ofΩbut sinceφis constant onAkit also has to be odd
there. This means thatP

π^{b}|Bπ^{b}|has to be odd. Hence, we also have thatP

π|Eπ∪Bπ|is the sum of
an even and an odd number and thus odd. So we conclude|C|+ 2|A|is odd and hence also|C|. Thus,
we find for anyAka completely labeledCπ_{k}.

1That is bearing exactly the label1, . . . , N−1onAk.

2 The boundary ofS is aσ-stable set so if it is partitioned by the labeling function intoAkwe know thatB_{N−1}(S) =
PK

k=11A_{k}BN−1(1A_{k}S)and by Lemma1.11 (iv)we can apply the induction hypothesis also onAk.

We defineS^{1} = PK

k=11A_{k}Cπ_{k} which by Remark 1.9is indeed a conditional simplex. Due toσ-
stability ofSit holdsS^{1} ⊆ S. By Remark1.12S^{1}has a diameter which is less then ^{N}_{N}^{−1}diam(S)and
sinceφis localS^{1}is completely labeled on wholeΩ.

The same argumentation holds for everym-fold barycentric subdivisionS^{m} ofS,m ∈ N, that is,
there exists a completely labeled conditional simplex in every m-fold barycentrically subdivided con-
ditional simplex which is properly labeled. Henceforth, subdividing S m-fold barycentrically and la-
bel it byφ^{m}: ext(S^{m}) → {1, . . . , N}(A), which is a labeling function as in Lemma 2.2, we al-
ways obtain a completely labeled conditional simplex S^{m+1} ⊆ S, form ∈ N. Moreover, sinceS^{1}
is completely labeled, it holdsS^{1} = PK

k=11_{A}_{k}Cπ_{k} as above whereCπ_{k} is completely labeled on A_{k}.
This means C_{π}_{k} = conv(Y_{1}^{k}, . . . , Y_{N}^{k}) with φ(Y_{j}^{k}) = j on A_{k} for every j = 1, . . . , N. Defining
V_{j}^{1} = PK

k=11_{A}_{k}Y_{j}^{k} for everyj = 1, . . . , N yieldsP({φ(V_{j}^{1}) = j}) = 1for every j = 1, . . . , N
and S^{1} = conv(V_{1}^{1}, . . . , V_{N}^{1}). The same holds for any m ∈ N and so that we can write S^{m} =
conv(V_{1}^{m}, . . . , V_{N}^{m})withP(φ^{m−1}({V_{j}^{m}) =j}) = 1for everyj= 1, . . . , N.

Now,(V_{1}^{m})_{m∈N}is a sequence in the sequentially closed,L^{0}-bounded setS, so that by [4, Corollary
3.9], there existsY ∈ S and a sequence(Mm)m∈N inN(A)such thatMm+1 > Mm for allm ∈ N
andlimm→∞V_{1}^{M}^{m} = Y P-almost surely. ForMm ∈ N(A),V_{1}^{M}^{m} is defined asP

n∈N1_{{M}_{m}_{=n}}V_{1}^{n}.
This means an element with index M_{m}, for some m ∈ N, equals V_{1}^{n} onA_{n},n ∈ N, where the sets
A_{n} are determined byM_{m}viaA_{n} = {M_{m} =n},n∈ N. Furthermore, asmgoes to∞,diam(S^{m})
is converging to zeroP-almost surely, and therefore it also follows thatlim_{m→∞}V_{k}^{M}^{m} =Y P-almost
surely for everyk = 1, . . . , N. Indeed, it holds|V_{k}^{m}−Y| ≤ diam(S^{m}) +|V_{1}^{m}−Y|for everyk =
1, . . . , N andm∈Nso we can use the sequence(Mm)m∈Nfor everyk= 1, . . . , N.

Let Y = PN

l=1αlXl and f(Y) = PN

l=1βlXl as well as V_{k}^{m} = PN

l=1λ^{m,k}_{l} Xl and f(V_{k}^{m}) =
PN

l=1µ^{m,k}_{l} X_{l} for m ∈ N. As f is local it holds that f(V_{1}^{M}^{m}) = P

n∈N1_{{M}_{m}_{=n}}f(V_{1}^{n}). By se-
quential continuity of f, it follows that lim_{m→∞}f(V_{k}^{M}^{n}) = f(Y)P-almost surely for every k =
1, . . . , N. In particular, lim_{m→∞}λ^{M}_{l} ^{m}^{,l} = α_{l} andlim_{m→∞}µ^{M}_{l} ^{m}^{,l} = β_{l} P-almost surely for every
l = 1, . . . , N. However, by construction,φ^{m−1}(V_{l}^{m}) =lfor everyl = 1, . . . , N, and from the choice
of φ^{m−1}, it follows that λ^{m,l}_{l} ≥ µ^{m,l}_{l} P-almost surely for every l = 1, . . . , N andm ∈ N. Hence,
α_{l}= lim_{m→∞}λ^{M}_{l} ^{m}^{,l}≥lim_{m→∞}µ^{M}_{l} ^{m}^{,l} =β_{l}P-almost surely for everyl= 1, . . . , N. This is possible
only ifα_{l}=β_{l}P-almost surely for everyl= 1, . . . , N, showing thatf(Y) =Y.

### 3 Applications

### 3.1 Fixed point theorem for sequentially closed and bounded sets in (L

^{0}

### )

^{d}

Proposition 3.1. LetKbe anL^{0}-convex, sequentially closed and bounded subset of(L^{0})^{d}andf:K →
Ka local, sequentially continuous function. Thenf has a fixed point.

Proof. Since K is bounded, there exists a conditional simplex S such that K ⊆ S. Now define the functionh:S → Kby

h(X) =

(X, ifX ∈ K, arg min{kX−Yk:Y ∈ K}, else.

This means, thathis the identity onKand the projection onKfor the elements inS \ K. Due to [4, Corollary 5.5] this minmium exists and is unique. Thereforehis well-defined.

We can characterizehby

Y =h(X)⇔ hX−Y, Z−Yi ≤0, for allZ∈ K. (3.1)

Indeed, lethX−Y, Z−Yi ≤0for allZ∈ K. Then
kX−Zk^{2}=k(X−Y) + (Y −Z)k

=kX−Yk^{2}+ 2hX−Y, Y −Zi+kY −Zk^{2}≥ kX−Yk^{2},
which shows the minimizing property of h. On the other hand, let Y = h(X). Since K is convex,
λZ+ (1−λ)Y ∈ Kfor anyλ∈(0,1](A)andZ ∈ K. By standard calculation,

kX−(λZ+ (1−λ)Y)k^{2}≥ kX−Yk^{2}

yields0 ≥ −2λhX,−Yi+ (2λ−λ^{2})hY, Yi+ 2λhX, Zi −λ^{2}kZk^{2}−2λ(1−λ)hZ, Yi. Dividing by
λ >0and lettingλ↓0afterwards yields

0≥ −2hX,−Yi+ 2hY, Yi+ 2hX, Zi −2hZ, Yi= 2hX−Y, Z−Yi, which is the desired claim.

Furthermore, for anyX, Y ∈ Sholds

kh(X)−h(Y)k ≤ kX−Yk. Indeed,

X−Y = (h(X)−h(Y)) +X−h(X) +h(Y)−Y =: (h(X)−h(Y)) +c which means

kX−Yk^{2}=kh(X)−h(Y)k^{2}+kck^{2}+ 2hc, h(X)−h(Y)i. (3.2)
Since

hc, h(X)−h(Y)i=−hX−h(X), h(Y)−h(X)i − hY −h(Y), h(X)−h(Y)i,

by (3.1), it follows thathc, h(X)−h(Y)i ≥ 0and (3.2) yieldskX−Yk^{2} ≥ kh(X)−h(Y)k^{2}. This
shows thathis sequentially continuous.

The functionf◦his a sequentially continuous function mapping fromStoK ⊆ S. Hence, there exists a fixed pointf ◦h(Z) =Z. Sincef ◦hmaps toK, thisZhas to be inK. But then we knowh(Z) =Z

and thereforef(Z) =Zwhich ends the proof.

Remark 3.2. In Drapeau, Jamneshan, Karliczek, and Kupper [5] the concept of conditional compactness
is introduced and it is shown that there is an equivalence between conditional compactness and conditional
closed- and boundedness in(L^{0})^{d}. In that context we can formulate the conditional Brouwer fixed point
theorem as follows. A sequentially continuous functionf:K → Ksuch thatKis a conditionally compact

andL^{0}-convex subset of(L^{0})^{d}has a fixed point.

### 3.2 Applications in Conditional Analysis on (L

^{0}

### )

^{d}

Working inR^{d}the Brouwer fixed point theorem can be used to prove several topological properties and
is even equivalent to some of them. In the theory of(L^{0})^{d}we will shown that the conditional Brouwer
fixed point theorem has several implications as well.

Define theunit ballin(L^{0})^{d}byB(d) ={X ∈(L^{0})^{d}: kXk ≤ 1}. Then by the former theorem any
local, sequentially continuous functionf:B(d)→ B(d)has a fixed point. Theunit sphereS(d−1)is
defined asS(d−1) ={X∈(L^{0})^{d}: kXk= 1}.