Choquet theory

The Choquet theory (after Gustave Choquet ) is a mathematical theory from the branch of functional analysis . It specifies the idea that the points of a compact , convex set of a locally convex space can be represented as "averaging" over the set of extreme points of this set.

The finite-dimensional case

Is a compact convex set of -dimensional real vector space , it can be any point from after the set of Minkowski as a convex combination of extreme points represent about . If the one-point measure denotes in , then follows for every affine mapping ${\displaystyle K\subset E}$ ${\displaystyle n}$ ${\displaystyle x_{0}}$ ${\displaystyle K}$ ${\displaystyle n+1}$ ${\displaystyle y_{0},\ldots ,y_{n}}$ ${\displaystyle x_{0}=\lambda _{0}y_{0}+\ldots +\lambda _{n}y_{n}}$ ${\displaystyle \varepsilon _{y_{i}}}$ ${\displaystyle y_{i}}$ ${\displaystyle a\colon K\mapsto \mathbb {R} }$

${\displaystyle a(x_{0})=\sum _{i=0}^{n}\lambda _{i}a(y_{i})=\int _{\mathrm {ex} (K)}a(x)\,\mathrm {d} \sum _{i=0}^{n}\lambda _{i}\varepsilon _{y_{i}}(x)}$,

where stands for the set of extremal points. The first equals sign follows from the given convex combination for and the affinity of , the second is clear, since the right-hand side is only a theoretical notation of the left-hand sum. In this sense, each point can be represented as an averaging with respect to a probability measure concentrated on the set of extreme points . The Choquet theory deals with infinitely dimensional generalizations of this fact. ${\displaystyle \mathrm {ex} (K)}$ ${\displaystyle x_{0}}$ ${\displaystyle a}$ ${\displaystyle K}$

general situation

Let it be a compact, convex set of a locally convex space. Then , the closure of the set of extremal points of , is also compact. ${\displaystyle K\subset E}$ ${\displaystyle {\overline {\mathrm {ex} (K)}}}$ ${\displaystyle K}$

Let it be the space of all continuous , affine mappings with values ​​in the real numbers. The restriction mapping into the algebra of continuous functions is isometric , as follows easily from the Kerin-Milman theorem, because it is a closed side, which in turn must contain extremal points of . can in this sense be understood as a subspace of . ${\displaystyle A(K)}$ ${\displaystyle K}$ ${\displaystyle \varphi \colon A(K)\rightarrow C\left({\overline {\mathrm {ex} (K)}}\right),a\mapsto a|_{\overline {\mathrm {ex} (K)}}}$ ${\displaystyle \{x\in K;\,a(x)=\sup a(K)\}}$ ${\displaystyle K}$ ${\displaystyle A(K)}$ ${\displaystyle C\left({\overline {\mathrm {ex} (K)}}\right)}$

Now be a point that we want to “average” over the set of extreme points. The mapping is a positive, continuous, linear functional on with norm 1 and, according to Hahn-Banach's theorem, can be continued in the same way as a continuous linear functional after . After the presentation set of Riesz-Markov therefore there is a regular Borel measure on so following formula: ${\displaystyle x_{0}\in K}$ ${\displaystyle a\mapsto a(x_{0})}$ ${\displaystyle A(K)}$ ${\displaystyle C({\overline {\mathrm {ex} (K)}})}$ ${\displaystyle {\overline {\mathrm {ex} (K)}}}$

${\displaystyle a(x_{0})=\int _{\overline {\mathrm {ex} (K)}}a(x)\,\mathrm {d} \mu (x)\qquad {\text{für alle }}a\in A(K)}$

This is also briefly noted as

${\displaystyle x_{0}=\int _{\overline {\mathrm {ex} (K)}}x\,\mathrm {d} \mu (x)}$,

which is supposed to mean nothing other than the preceding formula. In this case we say that the point is represented by the measure . In this sense, the Kerin-Milman theorem provides a kind of averaging over a point , the point results as an integral according to a measure at the end of the set of extreme points. ${\displaystyle x_{0}}$ ${\displaystyle \mu }$ ${\displaystyle x_{0}\in K}$ ${\displaystyle {\overline {\mathrm {ex} (K)}}}$ ${\displaystyle x_{0}}$

In many cases, the infinite-completion of the amount of all extreme points equal to the compact, convex set itself so that above statement is interesting, since one the as a measure then Einpunktmaß in can take. It would therefore be better if, as in the finite-dimensional case, one could do without the final formation, but the set of extremal points is generally not a Borel set, so that one cannot speak of Borel measures on this set. But if the compact convex set is even metrizable , then this case does not occur, and Choquet's theorem provides a representation of the desired type. In the non-metrizable case, one has to formulate differently because of the lack of measurability and comes to Bishop-de Leeuw's theorem . ${\displaystyle x_{0}}$

Choquet's theorem

If the compact, convex set is metrizable, then the above-mentioned measurability problems do not exist, because then the set of extreme points is a G _δ set and can therefore be measured by Borel.

• Theorem of Choquet (1956): Let be a metrizable, compact, convex set of a locally convex space and . Then there is a probability measure with vehicle in that represents the point . ${\displaystyle K}$ ${\displaystyle x_{0}\in K}$ ${\displaystyle \mu }$ ${\displaystyle {\mathrm {ex} (K)}}$ ${\displaystyle x_{0}}$

Bishop-de Leeuw's theorem

If the compact, convex set cannot be metrised, it can happen that the set of extreme points cannot be measured, and the statement that a measure has carriers in the set of extreme points does not make sense. One could weaken this condition by requiring that the measure vanishes on every Borel set that has an empty intersection with the set of extremal points. But even that turns out to be insufficient; you also have to reduce the borel quantities considered.

The Baire σ-ring , named after RL Baire , is understood to be the σ-ring generated by all compact G _δ sets . The elements of this ring are also called baire sets. ${\displaystyle \sigma }$

• Bishop - de Leeuw (1959) theorem : Let be a compact, convex set of a locally convex space and . Then there is a probability measure on which disjoint on each to the amount of extreme points Baire amount disappears and the point represents. ${\displaystyle K}$ ${\displaystyle x_{0}\in K}$ ${\displaystyle \mu }$ ${\displaystyle K}$ ${\displaystyle x_{0}}$

Remarks

The Bishop-de Leeuw theorem, also sometimes called the Choquet-Bishop-de Leeuw theorem, is a true generalization of the Choquet theorem, because in a compact, metrizable space every closed set is a compact set. ${\displaystyle G_{\delta }}$

Bishop-de Leeuw's theorem exacerbates Kerin-Milman's theorem, because the latter can easily be recovered from the former. If the point is a compact, convex set, then a representative measure from Bishop-de Leeuw's theorem obviously has the support at the end of the set of extremal points. By approximating the measure by discrete measures, one can see that the convex hull of is located at the end of , from which the Kerin-Milman theorem can easily be derived. ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle {\overline {\mathrm {ex} (K)}}}$

The theorems presented here have applications in the theory of Banach algebras , which then leads to the concept of the Choquet margin , and also in other areas of functional analysis. For further details, see the textbook by RR Phelps given below.

Individual evidence

1. ^ E. Bishop, K. de Leeuw: The representation of linear functionals by measures on sets of extreme points , Ann. Inst. Fourier (Grenoble) 1959, Volume 9, Pages 305-331, see page 327
2. ^ RR Phelps: Lectures on Choquet's Theorem , van Nostrand (1966), Proposition 1.3
3. ^ RR Phelps: Lectures on Choquet's Theorem , van Nostrand (1966), Theorem in Chapter 3
4. ^ RR Phelps: Lectures on Choquet's Theorem , van Nostrand (1966), Theorem in Chapter 4