Question

Let \(K=\overline{\operatorname{conv}}\left\\{\\{0\\} \cup\left\\{e_{n}\right\\}_{n=1}^{\infty}\right\\} \subset \ell_{2}\), where \(e_{n}\) are the standard unit vectors. Show that there does not exist a probability measure \(\mu\) on \(K\) supported by \(\left\\{e_{n}\right\\}\) that represents \(0 .\) This shows that Choquet's representation theorem cannot give a measure supported by the strongly exposed points \(\left\\{e_{n}\right\\}\) of \(K\)

Short answer

No such probability measure can represent \(0\) because it requires all weighting coefficients to be \(0\), which contradicts the total measure condition.

Step-by-step solution

Understand the Problem

We are given a set \(K\) in the Hilbert space \(\ell_2\) which is the closed convex hull of the set \(\{0\} \cup \{e_{n}\}_{n=1}^{\infty}\), where \(e_{n}\) are the standard unit vectors in \(\ell_2\). We need to show that there does not exist a probability measure supported on \(\{e_{n}\}\) that can represent the point \(0\).

Define the Set \(K\)

The set \(K = \overline{\operatorname{conv}}\{0 \cup \{e_{n}\}_{n=1}^{\infty}\}\) consists of all points in \(\ell_2\) that are convex combinations of the point \(0\) and the sequence of unit vectors \(\{e_{n}\}\) along with their limit points.

Convex Combination Properties

Recall that a convex combination \(\sum_{k} \alpha_{k} x_{k}\) of points \(x_{k}\) with coefficients \(\alpha_{k} \geq 0\) and \(\sum_{k} \alpha_{k} = 1\) produces a point within the convex hull. By the definition of \(K\), any point in \(K\) can be written as such a sum involving \(0\) and \(\{e_n\}\).

Probability Measure Representation

A probability measure \(\mu\) on \(K\) supported by \(\{e_n\}\) means \(\mu\) is a weighted sum of the Dirac measures \(\delta_{e_n}\). To represent \(0\), we need to express \(0\) as the 'expectation' or weighted average of \(e_n\), i.e., \(0 = \sum_{n=1}^{\infty} \mu(\{e_n\}) e_n\) where \(\mu(\{e_n\}) \geq 0\) and \( \sum_{n=1}^\infty\mu(\{e_n\}) = 1.\)

Contradiction

If \(0\) were to be represented as a weighted average of \(e_n\), each \(e_n\) being a standard unit vector, we would require \(\mu(\{e_n\}) e_n = 0\) for all \(n\). This implies all \(\mu(\{e_n\}) = 0\) since \(e_n\) are non-zero vectors. However, this would violate the condition that \(\sum_{n=1}^{\infty} \mu(\{e_n\}) = 1\). Therefore, no such measure \(\mu\) exists.

Key concepts

Convex Hulls

The convex hull of a set of points is a fundamental concept in functional analysis and geometry. It is the smallest convex set that contains all the given points. The convex hull can be visualized as the shape formed by stretching a rubber band around the outside of these points. Mathematically, if we have a set of points \(\text{{P}} = \{x_{1}, x_{2}, \ldots, x_{n} \}\), then the convex hull is defined as the set of all convex combinations \(\text{conv}(P) = \left\{ \sum_{i=1}^{n} \lambda_{i}x_{i} \mid \lambda_{i} \geq 0, \sum_{i=1}^{n} \lambda_{i} = 1 \} \)\).
This means any point in the convex hull is a combination of the original points, where the coefficients \(\lambda_i\) are non-negative and sum up to 1. For the set \(K\) in our original exercise, we have \(K = \overline{\operatorname{conv}}\{\{0\} \cup \{e_{n}\}_{n=1}^{\infty}\}\), which includes the origin and all points that can be formed by convex combinations of \( \{e_n\} \inite sequence.
Understanding convex hulls helps us comprehend the structure of \(K\) and why certain properties hold true.

Probability Measures

Probability measures are essential in understanding distributions in functional spaces. A probability measure \(\mu\) on a set assigns a probability to each possible event in a way that the total measure of the complete set is 1. This concept can be extended to sets of points in spaces like \( \ell_2 \) spaces.
In the context of the original problem, a probability measure \(\mu\) supported by the set \( \{e_{n}\}_{n=1}^{\infty} \) implies that \(\mu \) is a weighted sum of point masses at \(\{e_{n}\}\). To represent the origin using this measure, we would need to express \(0 \) as an 'expectation' or weighted average of the points \(\{e_{n}\}\). This requires \(\sum_{n=1}^{\infty} \mu(\{e_{n}\}) e_{n} = 0 \), with each weight \( \mu(\{e_{n}\}) \geq 0 \) and summing up to 1.
This fundamental concept in probability helps in arguing the impossibility of representing 0 using the given measure in the exercise, leading to the contradiction we sought.

Hilbert Spaces

Hilbert spaces are complete inner-product spaces that generalize the notion of Euclidean space. They are fundamental in functional analysis, providing a setting for discussing convergence, continuity, and orthogonality.
In finite-dimensional space, each Hilbert space can be thought of as a vector space with a dot product. This dot product allows us to measure angles and lengths. In an infinite-dimensional setting, like the space \( \ell_2 \) mentioned in the problem, the inner product is defined as \(\langle x, y \rangle = \sum_{i=1}^{\infty} x_i y_i \).
Since \( \ell_2 \) spaces consist of all sequences whose series of squared entries converges, understanding these spaces helps us see why certain points can't be represented in specific ways using measures supported on countable sets of unit vectors.

Choquet's Theorem

Choquet's Theorem is a significant result in convex analysis and functional analysis. It states that under certain conditions, any point in a compact convex set can be represented as an integral (or convex combination) of extreme points of the set.
In the context of our problem, Choquet's Theorem would suggest that for the set \(K = \overline{\operatorname{conv}}\{\{0\} \cup \{e_{n}\}_{n=1}^{\infty}\}\), any point, including the origin, should be representable as a combination of points from the set of extreme points \(\left\{e_{n}\}\right\}\).
However, the exercise shows that this is not always possible. Specifically, it demonstrates that there doesn't exist a probability measure supported by \(\left\{ \{e_{n} \}\right\}\) that can represent the origin. This limitation is crucial because it highlights the nuanced conditions under which Choquet's Theorem applies, emphasizing that not all convex combinations or measures produce expected representations.