ARpDq For a compact convex subset D of an lctvs E, represents the set of all real-valued affine functions on D. SpDq For a compact convex subset D of an lctvs E, SpDq represents the set of all continuous convex functions on D.

## Carath´ eodory’s theorem

Let M be a proper subspace of an lctvs E and p a sublinear functional on E. Let f be a linear functional on M such that Re fpxq ďppxq for all xPM. Thus, by the 2nd principle of mathematical induction, the result is true for any lctvs of dimension n, for any PN.

## Resultant of a Measure

Then there exists a unique xPX such that x “rpµq and the mapping r:PpYq ÝÑX is defined as µÞÝÑrpµq w˚-connected. Now, since X is compact, there exists a convergent subnet, say pxαiq, such that xαi ÝÑx, for some xPX.

## An application of Krein-Milman theorem

To show that K is bounded, it suffices to show from (iii) that for every m P N and n P NY t0u, suptpm,npfq : f P Ku is finite, i.e. The endpoints of K are those functions f functions of the form for each αP r0.8s, for all x±0, fpxq “ e´αx.

## Preliminaries of Affine functions

Then we can show on similar lines using Example 3.1.2 that D is a w˚-compact convex subset of `2. 31 Chapter 3 Choquet's Theorem for Metric Compact Convex Sets It is clear that for all m, n P N, fmn P CRpKq and annihilates outside the open sphere Bpxn,m1q.

## Main results: Choquet-Bishop-De Leeuw exis- tence Theorem

We now show that A forms a separable subalgebra of CRpKq, so by the Stone-Weierstrass theorem it follows that A is dense in CRpKq. The strict inequality in the third step is due to the fact that the mapping t Ñ t2 is strictly convex on R. Therefore, with the Hahn Banach extension theorem, we can extend Φ to the entire spaceCRpDq, so that for each f PCRpDq, Φpfq ď ppfq holds.

33 Chapter 3 Choquet's Theorem for Metrizable Compact Convex Sets By the Riesz Representation Theorem, there exists a regular Borel measure of D such that for each f PCRpDq, Φpfq “ş. Let us recall that if X is a separable normed linear space, then the double sphere is a metrizable compact convex subset of pX˚, w˚q (see Appendix B), hence Theorem 3.2.1 is applicable to such spaces. Now, by Choquet's theorem for metrizable compact convex sets, for every x˚P BX˚, there exists µPPpBX˚qsupported on extpBX˚q such that x˚ "rpµq.

From our previous arguments, it is clear that PpβNq and thus BMpβNq are some examples of compact convex subsets which are not measurable. Let K be a compact Hausdorff space and U be a sublattice of CRpKq such that for any f PCRpKq, for any two points x, y PK and for any positive numberε there exists a function fxy PU such that|fpxq ´fxypxq| ăε,|fpyq ´fxypyq| ă ε.

## A new setup: Boundary measure

## Some applications

Consequently, there exists µ P PpUq such that g “ rpµq, support of µ is contained in extpDq and also for every weak˚ continuous affine function Φ on U,. Let E be a real Banach space and K a weakly compact convex subset of E˚ such that extpKq is norm separable. 50 Chapter 4 Choquet's theorem for nonmetrizable compact convex sets xk PU since Spνkq Ăsupppgkq ĂU.

51 Chapter 4 Choquet's theorem for nonmetrizable compact convex sets is an algebra, it follows that ν, λP AK. It follows that ν is absolutely continuous with respect to λ`µ and therefore the Radon-Nikodym derivative has say s. Let X be a non-empty compact convex set in an lctvsE such that X is contained in a hyperplane which does not contains zero.

Let λ P M and µ P M`pXq be such that µ ĺ λ (where ĺ is the order . 57 Chapter 5 Choquet Simplex and its characteristics induced by M`pXq). Otherwise, without loss of generality, we can remove vectors from the set txiuni“1 such that txiumi“1, where 1ďmăn is a linearly independent set.

## Definitions and basic properties

Adding a large positive constant M to both sides of the above inequality, such that g “f`M ě0, we obtain. Let M be the subspace CRpKq (or CCpKq) that separates the points of K and contains constant functions. Then k P BpMq if and only if µ “ δk is the only probability measure on K such that fpkq “ş.

Thus, there exist two separate measures µ1 and µ2 on K representing φpkq, such that for any fPM, φpkqpfq “ µ12pfq ` µ22pfq. If we show that there exists fP M such that sup|fpKzUq| ă sup|fpUq|, then we get a contradiction with the fact that B is a limit for M. By definition of weak˚ topology and the fact that the dual ofpM˚, w˚qisM itself, there exists g1, g2.

Let M be a subspace CRpKq (or CCpKq) such that it separates the points of K and contains constant functions. By the extension theorem of Hahn Banach, L can be expanded to a continuous linear functional ˜L on CRpKq such that }L} “ }L}.

## Choquet Boundary for uniform algebras

Then paq implies that tgnu is Cauchy in A, and since A is complete, there exists f P A such that gn Ñf is uniform. If K is metrizable compact Hausdorff space and A is a function algebra in CCpKq, then the Choquet limit BpAq coincides with the set of vertices of A. We claim that the Choquet limit of ApDq coincides with its ˘Silov limit, and these is equal to the limit T " tz PC: |z| “ 1u of D.

It follows from the maximum modulus principle that the Choquet limit of BpMq does not contain 1, since BpMq and the vertex set are the same for this case. A point xP S is a smooth point of the unit sphere E if there exists a unique f PE˚ such that fpxq “ 1“ }f}8. According to the Banach Alaoglu theorem, there exist convergent subnetworks tfku and tgku, say tfαu and tgαu, respectively, such that fα ÝwÝÑ˚ f and gα ÝwÝÑ˚ g.

Let P be a set y P K such that there exists a smooth point f of the unit sphere M such that fpyq “ }f}8. Suppose that KpMq is not a w˚-closed convex hull of φpPq, then there exists L P KpMqzconvw˚pφpPqq.

## Choquet Boundary and approximation theory

Let M be a subset of CRpKq such that for any countable family of positive operators pTnqnPN, whenever for each g PM,Tng Ñg (ie, Tng converges to uniformly), we have Tnf Ñf for each f PCRpKq. M is a Korovkin set CRpKq if and only if the linear range of M is a Korovkin set CRpKq. Let K be a metrizable compact space and let M be a linear subspace in CRpKq that contains 1 and separates the points of K.

Then M is a Korovkin group iCRpKq if and only if the Choquet limit BpMq for M is all K. To prove BpMq " K, using Theo-rem6.1.4, it suffices to prove that for every PK, μ"δt is the only probability measure onK such that µpgq “gptq, for all PM. Conversely, let tTnunPN be a family of positive operators on CRpKq such that for every g P M, Tng Ñ g as n Ñ 8.

For the simplicity of notation, assume that}Tnf ´f}8u is the initial subsequence and for eachP N, choosexnP Ksodat}Tnf´f}8 “ |Tnfpxnq ´fpxnq|. It is clear that Ln is a linear functional on CRpKq and also |Lnpgq| ď }Tn}}g}8, which implies that Ln is bounded.

## Convexity in a topological vector space

Characterizations

Extreme set(or Face)

## Smoothness in a Normed Linear Space

Now, clearly, pp´yq “ ´qpyq, for any y P Rn and thus q is a superlinear functional on Rn. Therefore, Φ can be extended by Hahn Banach theorem to Rn, which is also by p. Now for some y P Rn, suppose ppyq ą t1 ‰ t2 ą qpyq, then there exist two distinct linear functionals on spantyu and finally by Hahn Banach theorem, two distinct linear functionals on Rn, contradicting the uniqueness of the given Λ .

A normed linear space X is said to be smooth at x0 P Rn if there exist unique x˚P SX˚ such that x˚px0q “ }x0}. The other concepts such as Fr´echet differentiability, Uniform Gˆateaux differentiability, Uniform Fr´echet differentiability etc. are all strengthenings of Definition A.2.3. All these concepts are equivalent under the assumption that the unit sphere is norm compact.

Then there exists a unique topology τ oneE for which B is a neighborhood basis of 0 and τ is the smallest such topology in E such that the functions on Σ are continuous. With these notations defined above, the topology defined in X˚˚ is said to be a weak topology˚ if Σ “ X˚ and E “ X˚˚ and is denoted by σpX˚˚, X˚q.

## Basic Properties

### Banach-Alaoglu Theorem

It is well known in the infinite-dimensional case that the norm topology cannot allow the closed unit ball to be compact, but the situation can occur in a weaker topology. It is clear that a map is not well defined since the series on the RHS is uniformly convergent by the Weierstrass-M test. A measure µ is said to be regular if it is both inner and outer regular.

Let K be any compact Hausdorff space and MpKq the space of all regular measures of the Borel Complex in K.

## Riesz Representation Theorem

87 Appendix C On some basic results in Measure Theory unique complex (signed) measures µ inpK,Mq such that for all f PCpKq,.

## Probability measure

It is due to Birkhoff that the condition D.1 is equivalent to px^yq _z “ px_zq ^ py_zq. IfpL,ďq is a lattice, the mappingspx, yq ÞÑx_yandpx, yq ÞÑ x^y are usually called the lattice operations. As laws of composition they are idempotent, associative, commutative and satisfix^ px_yq “xasook x_ px^yq “x.

On the other hand, it is not difficult to verify that if the nonempty set L is equipped with two composition laws with these properties, then x ě y if and only if x_y “ y defines an order under which L is a lattice in the sense of definition D.0.1 . Also recall that a network L is called (countable) complete if every (countable) subset of L has a minimum upper bound and a maximum lower bound. L0 is called a (countably) perfect sublattice of L if L0 is closed with respect to the formation of arbitrary (countably) infimes and supremes.

However, the infima or suprema subnets may not be the same as those in the network. The set LM of all mappings f : M Ñ L is a lattice under canonical order defined by:f ď g if and only if fptq ď gptq for all tPM.

## Basic Properties

We can easily use induction on the number of elements in I and in J and reduce the proof to the case I “J” t1,2u. Alfsen, Compact convex sets and boundary integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57. 4] Michel Grabisch, Ivan Kojadinovic, Patrick Meyer, A review of methods for capacity identification in Choquet integral based multi-attribution utility theory: Applications of the Kappalab R Package European J .

6] Michel Grabisch and Christophe Labreuche, A decade of application of Choquet and Sugeno integrals in multi-criteria decision aids, Annals of Operations Research.