Brouwer fixed point theorem in
© Drapeau et al.; licensee Springer. 2013
Received: 17 May 2013
Accepted: 21 October 2013
Published: 19 November 2013
The classical Brouwer fixed point theorem states that in every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, let be the set of random variables. We consider as an -module and show that local, sequentially continuous functions on -convex, closed and bounded subsets have a fixed point which is measurable by construction.
MSC:47H10, 13C13, 46A19, 60H25.
Keywordsconditional simplex fixed points in
The Brouwer fixed point theorem states that a continuous function from a compact and convex set in 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 et al. , inspired by the theory developed by Filipović et al.  and Guo , studied as an -module, discussing concepts like linear independence, σ-stability, locality and -convexity. Based on this, we define affine independence and conditional simplexes in . 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 suitable σ-combination of elements of the barycentric subdivision along a partition of Ω. 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 construction. Hence, even though we follow the constructions and methods used in the proof of the classical result in (cf. ), we do not need any measurable selection argument.
is a random variable for any fixed ,
is a continuous function for any fixed .
For R there exists a random fixed point which is a random variable such that for any ω (cf. [5–7]). In contrast to this ω-wise consideration, our approach is completely within the theory of . All objects and properties are therefore defined in that language and proofs are done with -methods. Moreover, the connection between continuous random operators on and sequentially continuous functions on is not entirely clear.
An application, though not studied in this paper, is for instance possible in economic theory or optimization in the context of . Therein the methods from convex analysis are used to obtain equilibrium results for translation invariant utility functionals on . 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 to .
The present paper is organized as follows. In the first section, we present the basic concepts concerning as an -module. We define conditional simplexes and examine their basic properties. In the second section, 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 section, we show a fixed point result for -convex, bounded and sequentially closed sets in . 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 and, on the other hand, an intermediate value theorem in .
1 Conditional simplex
For a probability space , let be the space of all -measurable random variables, where P-almost surely equal random variables are identified. In particular, for , the relations and have to be understood P-almost surely. The set with the P-almost everywhere order is a lattice ordered ring, and for a non-empty subset , we denote the least upper bound by and the greatest lower bound by , respectively (cf. ). For , we denote the constant random variable by m. Further, we define the sets , and . The set of random variables with values in a set is denoted by . For example, is the set of -measurable functions with values in , and .
An element such that for all is called a strict convex combination of . Moreover, a set is said to be -convex if for any and , it holds that .
We call bounded if and sequentially closed if it contains all P-almost sure limits of sequences in . Further, the diameter of is defined as .
Indeed, first we show that (1.1) implies (1.2). Let and . Then . By assumption (1.1), it holds that , thus also . To see that (1.2) implies (1.1), let . With , it holds that . By assumption (1.2), .
Remark 1.2 We observe that if are affinely independent, then for and for are affinely independent. Moreover, if a family is affinely independent, then also are affinely independent on , which means from and it always follows that for all .
such that are affinely independent. We call the dimension of .
Indeed, since and , it follows from (1.2) that for all .
Remark 1.5 Note that the present setting - -modules and the sequential P-almost sure convergence - is of local nature. This is, for instance, not the case for subsets of or the convergence in the -norm for . First, is not closed under multiplication and hence neither a ring nor a module over itself, so that we cannot even speak about affine independence. Second, it is in general not a σ-stable subspace of . However, for a conditional simplex in such that any is in , it holds that is uniformly bounded by . This uniform boundedness yields that any P-almost sure converging sequence in is also converging in the -norm for due to the dominated convergence theorem. This shows how one can translate results from to .
Since a conditional simplex is a convex hull, it is in particular σ-stable. In contrast to a simplex in , the representation of as a convex hull of affinely independent elements is unique but up to σ-stability.
Proposition 1.6 Let and be families in with . Then . Moreover, are affinely independent if and only if are affinely independent.
If is a conditional simplex such that , then it holds that .
Therefore, and the reverse inclusion holds analogously.
Hence, it holds that . It follows that . This is a contradiction to the former part of the proof (because ).
Next, we show that in a conditional simplex it holds that X is in if and only if there do not exist Y and Z in and such that . Consider which is for a partition . Now assume to the contrary that we find and in such that . This means that . Due to uniqueness of the coefficients (cf. (1.3)) in a conditional simplex, we have for all . By means of , it holds that if and only if . Since the last equality holds for all k, it follows that . Therefore, we cannot find Y and Z in such that X is a strict convex combination of them. On the other hand, consider such that . This means such that there exist and and with on B and on B. Define . Then define if and , , and . Thus, and fulfill but both are not equal to X by construction. Hence, X can be written as a strict convex combination of elements in . To conclude, consider . Since , it is not a strict convex combination of elements in , in particular, of elements in . Therefore, X is also in . Hence, . With the same argumentation, the other inclusion follows. □
As an example, let us consider . For an arbitrary , it holds that and are affinely independent and . Thus, the conditional simplex can be written as a convex combination of different affinely independent elements of . This is due to the fact that for all .
Remark 1.7 In , let be the random variable which is 1 in the i th component and 0 in any other. Then the family is affinely independent and . Hence, the maximal number of affinely independent elements in is .
The characterization of leads to the following definition.
Definition 1.8 Let be a conditional simplex. We define the set of extremal points . For an index set I and a collection of conditional simplexes, we denote .
shows . The other inclusion follows by considering and defining . To show that are affinely independent, we consider . Then by (1.4) it holds that and since is a conditional simplex, for all and . From the fact that is a partition, it follows that for all .
We will prove the Brouwer fixed point theorem in the present setting using an -module version of Sperner’s lemma. As in the unconditional case, we have to subdivide a conditional simplex into 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.
We call the barycentric subdivision of .
has dimension N, .
is a conditional simplex of dimension and for , .
For , let . All conditional simplexes , , of dimension s subdivide barycentrically.
with . Since , the affine independence of is obtained by the affine independence of . Therefore all are conditional simplexes.
which shows that on . Condition (ii) is fulfilled by construction.
which is only possible if since . Furthermore, if is of dimension N, by (1.5) it follows that . This shows (iii).
Further, for , the elements of dimension s are exactly the ones with . To this end, let be of dimension s. This means there exists an element Y in this intersection such that with for all and for . As an element of , this Y has a representation of the form for and for every . Suppose now that there exists some with . Then due to and the uniqueness of the coefficients (cf. (1.3)) in a conditional simplex, it holds that and within for all . This means and hence Y is the convex combination of elements with . This contradicts the property that for s elements. Therefore, is exactly the barycentric subdivision of , which has been shown to fulfill the properties (i)-(iii). □
Subdividing a conditional simplex barycentrically, we obtain . Dividing every barycentrically results in a new collection of conditional simplexes and we call this the two-fold barycentric subdivision of . Inductively, we can subdivide every conditional simplex of the th step barycentrically and call the resulting collection of conditional simplexes the m-fold barycentric subdivision of and denote it by . Further, we define to be the σ-stable hull of all extremal points of the conditional simplexes of the m-fold barycentric subdivision of . Notice that this is the σ-stable hull of only finitely many elements, since there are only finitely many simplexes in the subdivision, each of which is the convex hull of N elements.
Since this holds for any , it follows that the diameter of , which is an arbitrary conditional simplex of the m-fold barycentric subdivision of , fulfills . Since and , for , it follows that for for every sequence .
2 Brouwer fixed point theorem for conditional simplexes
for all .
for all ,
where and are determined by and . Then ϕ is a proper labeling function.
Moreover, the set of functions fulfilling these properties is non-empty.
Assume, to the contrary, that on for all with on A. Then it holds that on A, which yields a contradiction. Thus, ϕ is a labeling function. Moreover, due to (i), it holds in particular that , which shows that ϕ is proper.
To prove the existence for with , , let , . Then we define the function ϕ at X as , . It has been shown that ϕ maps to and is proper. It remains to show that ϕ is local. To this end, consider , where and . Due to uniqueness of the coefficients in a conditional simplex, it holds that , and due to locality of f, it follows that . Therefore it holds that . Hence, on . On the other hand, we see that is i on any , hence it is i on . Thus, , which shows that ϕ is local. □
If we label Y by the rule explained in Lemma 2.2, ϕ takes the values and . Therefore, we can really distinguish on which sets . Yet, using a deterministic labeling of Y, we would lose this information.
Theorem 2.3 Let be a conditional simplex in . Let further be a local, sequentially continuous function. Then there exists such that .
Proof We consider the barycentric subdivision of and a proper labeling function ϕ on . First, we show that we can find a completely labeled conditional simplex in . By induction on the dimension of , we show that there exists a partition such that on any there is an odd number of completely labeled . The case is clear since a point can be labeled with the constant index 1 only.
Suppose that the case is proven. Since the number of of the barycentric subdivision is finite and ϕ can only take finitely many values, it holds for all that there exists a partition , , where is constant on any . Therefore, we find a partition such that on is constant for all V and . Fix now.
In the following, we denote by those conditional simplexes for which are -dimensional (cf. Lemma 1.11(iv)), therefore . Further we denote by these conditional simplexes which are not of the type , that is, . If we use , we mean a conditional simplex of arbitrary type. We define:
to be the set of which are completely labeled on .
to be the set of P-almost completely labeled , that is,
to be the set of intersections which are -dimensional and completely labeled on .b
to be the set of intersections which are completely labeled on .
It holds that and hence . Since is at most -dimensional, it holds that and hence . Moreover, we know that is -dimensional on if and only if this holds on the whole Ω (cf. Lemma 1.11(iii)) and on if and only if this also holds on the whole Ω (cf. Lemma 1.11(iv)). So, it does not play any role if we look at these sets which are intersections on or on Ω since they are exactly the same sets.
If , then and if , then . If , then and if , then . Therefore it holds that .
If we pick , we know that there always exists exactly one other such that (Lemma 1.11(iii)). Therefore is even. Moreover, subdivides barycentrically, and hence we can apply the inductive hypothesis (on ). Indeed, the set is a σ-stable set, so if it is partitioned by the labeling function into , we know that and by Lemma 1.11(iv) we can apply the induction hypothesis also to every , . Thus, the number of completely labeled conditional simplexes is odd on a partition of Ω, but since ϕ is constant on , it also has to be odd there. This means that has to be odd. Hence, we also have that is the sum of an even and an odd number and thus odd. So, we conclude is odd and hence also . Thus, we find for any a completely labeled .
We define which by Remark 1.9 is indeed a conditional simplex. Due to σ-stability of , it holds that . By Remark 1.12, has a diameter which is less than and since ϕ is local, is completely labeled on the whole Ω.
The same argumentation holds for every m-fold barycentric subdivision of , , that is, there exists a completely labeled conditional simplex in every m-fold barycentrically subdivided conditional simplex which is properly labeled. Henceforth, subdividing m-fold barycentrically and labeling it by , which is a labeling function as in Lemma 2.2, we always obtain a completely labeled conditional simplex for . Moreover, since is completely labeled, it holds as above, where is completely labeled on . This means with on for every . Defining for every yields for every and . The same holds for any and so that we can write with for every .
Now, is a sequence in the sequentially closed, -bounded set , so that by [, Corollary 3.9], there exists and a sequence in such that for all and P-almost surely. For , is defined as . This means an element with index , for some , equals on , , where the sets are determined by via , . Furthermore, as m goes to ∞, is converging to zero P-almost surely, and therefore it also follows that P-almost surely for every . Indeed, it holds that for every and , so we can use the sequence for every .
Let and as well as and for . As f is local, it holds that . By sequential continuity of f, it follows that P-almost surely for every . In particular, and P-almost surely for every . However, by construction, for every , and from the choice of , it follows that P-almost surely for every and . Hence, P-almost surely for every . This is possible only if P-almost surely for every , showing that . □
3.1 Fixed point theorem for sequentially closed and bounded sets in
Proposition 3.1 Let be an -convex, sequentially closed and bounded subset of , and let be a local, sequentially continuous function. Then f has a fixed point.
This means, that h is the identity function on and the projection on for the elements in . Due to [, Corollary 4.5] this minimum exists and is unique. Therefore h is well defined.
which is the desired claim.
by (3.1), it follows that . Therefore, by (3.2). This shows that h is sequentially continuous.
The function is a sequentially continuous function mapping from to . Hence, there exists a fixed point . Since maps into , this Z has to be in . But then we know and therefore , which ends the proof. □
Remark 3.2 In Drapeau et al.  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 . In that context we can formulate the conditional Brouwer fixed point theorem as follows. A sequentially continuous function such that is a conditionally compact and -convex subset of has a fixed point.
3.2 Applications in conditional analysis on
Working in , 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 , we will show that the conditional Brouwer fixed point theorem has several implications as well.
Define the unit ball in by . Then, by the former theorem, any local, sequentially continuous function has a fixed point. The unit sphere is defined as .
Definition 3.3 Let and be subsets of . An -homotopy of two local, sequentially continuous functions is a jointly local, sequentially continuous function such that and . Jointly local means for any partition , in and in . Sequential continuity of H is therefore whenever and both P-almost surely for and .
Lemma 3.4 The identity function of the sphere is not -homotopic to a constant function.
The proof is a consequence of the following lemma.
Lemma 3.5 There does not exist a local, sequentially continuous function which is the identity on .
Proof Suppose that there is this local, sequentially continuous function f. Define the function by . Then the composition , which actually maps to , is local and sequentially continuous. Therefore, this has a fixed point Y which has to be in since this is the image of . But we know and and hence . Therefore, Y cannot be a fixed point (since ), which is a contradiction. □
It directly follows that the identity on the sphere is not -homotopic to a constant function. In the case , we get the following result which is the -module version of an intermediate value theorem.
Lemma 3.6 Let with and . Let further be a local, sequentially continuous function and . If Y is in , then there exists with .
Proof Since f is local, it is sufficient to prove the case for which is . For the general case, we would consider A and separately, obtain , and by locality we have . So, suppose that Y is in in the rest of the proof.
Notice that as a sum, product, and composition of local, sequentially continuous functions, g is so as well. Hence, g has a fixed point . If , it must hold that , which means , which is a contradiction. If , it follows that , which is also a contradiction. Hence, , which means .
If on B and on C, then on . Then we find such that on D. In total . This shows the claim for general Y in . □
Let , . This finite collection of measurable sets fulfills . We can construct a partition such that for some and for all . Such a partition fulfills the required property.
That is bearing exactly the label on .
We thank Asgar Jamneshan for fruitful discussions. The first author was supported by MATHEON project E11. The second author was supported by Konsul Karl und Dr. Gabriele Sandmann Stiftung. The fourth author was supported by Evangelisches Studienwerk Villigst.
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