Question

Show that \(\overline{\operatorname{conv}}\left(\operatorname{Ext}\left(B_{C[0,1]^{*}}\right)\right) \neq B_{C[0,1]^{*}}\) Hint: Show that the Lebesgue measure \(\lambda\) is not in \(\overline{\operatorname{conv}}\left(\operatorname{Ext}\left(B_{C[0,1] *}\right)\right)\). Indeed, if \(\lambda\) is norm-close to \(\sum \alpha_{i} \delta_{t_{i}}\), find \(f \in B_{C[0,1]}\) such that \(f\left(t_{i}\right)=0\) but \(\lambda(f)\) is close to 1 .

Short answer

\(\lambda\) is not in \(\bar{\text{conv}}(\text{Ext}(B_{C[0,1]^{*}}))\) because it cannot be approximated by finite convex combinations of Dirac measures.

Step-by-step solution

Understand the Sets and Definitions

To start, let's understand the sets involved. The set \(\text{Ext}(B_{C[0,1]^{*}})\) denotes the extreme points of the unit ball of the dual space \(C[0,1]^{*}\), also known as the space of Radon measures on \([0,1]\). The set \(\text{conv}(\text{Ext}(B_{C[0,1]^{*}}))\) represents the convex hull of these extreme points, and \(\bar{\text{conv}}\) indicates the closure of this convex hull in the norm topology.

Identify the Structure of Extreme Points

The extreme points of the unit ball in \(C[0,1]^{*}\) are the Dirac measures \(\delta_t\) at points \(t \in [0,1]\). Thus, \(\text{conv}(\text{Ext}(B_{C[0,1]^{*}}))\) consists of finite convex combinations of Dirac measures, which represent discrete measures.

Consider Integration Against Lebesgue Measure

The Lebesgue measure \(\lambda\) on \([0,1]\) is a continuous measure and is not a finite convex combination of Dirac measures. Suppose \(\lambda\) were in the closure, \(\bar{\text{conv}}(\text{Ext}(B_{C[0,1]^{*}}))\). This would imply that \(\lambda\) can be approximated in norm by elements in \(\text{conv}(\text{Ext}(B_{C[0,1]^{*}}))\).

Formulate an Approximation Argument

If \(\lambda\) can be approximated by \(\sum \alpha_i \delta_{t_i}\), find a function \(f \in B_{C[0,1]}\) such that \(f(t_i) = 0\). Specifically, the function is chosen so that \(\lambda(f) \approx 1\), while the approximation sum suggests \(\sum \alpha_i f(t_i) = 0\).

Choose an Appropriate Function

Choose \(f\) to be a continuous function from \([0,1]\) to \([0,1]\) which is zero at all \(t_i\) and integrates to approximately 1 with respect to \(\lambda\). This function exists because for any finite set of points, it's possible to construct such a function given the properties of the Lebesgue integral.

Reach a Contradiction

We have \(\lambda(f) \approx 1\) by construction, but at the same time \(\sum \alpha_i \delta_{t_i}(f) = 0\). Thus, \(\lambda\) is norm-separated from any finite convex combination of Dirac measures, showing that \(\lambda\) is not in \(\bar{\text{conv}}(\text{Ext}(B_{C[0,1]^{*}}))\).

Conclusion

Since \(\lambda\) is in \(B_{C[0,1]^{*}}\) but not in \(\bar{\text{conv}}(\text{Ext}(B_{C[0,1]^{*}}))\), it follows that \(\bar{\text{conv}}(\text{Ext}(B_{C[0,1]^{*}})) eq B_{C[0,1]^{*}}\).

Key concepts

Extremal Points

When we talk about extremal points, we refer to points in a convex set that cannot be written as a combination of other points in that set. Think of them as the 'building blocks' of the set.
In the unit ball of the dual space \(\text{C}[0,1]^{*}\), the extremal points are particularly significant.
These are the Dirac measures, \(\delta_t\), where \(t \in [0,1]\).
Dirac measures are very simple—they just 'pick out' the value of a function at a single point.
In summary, extremal points give us a fundamental way to describe more complex sets and structures.

Convex Hull

A convex hull is the smallest convex set that includes a given set of points.
Imagine stretching a rubber band around a bunch of pins; the shape that the band forms is like the convex hull of those pins.
In our context, \(\text{conv}(\text{Ext}(B_{C[0,1]^{*}}))\) represents combinations of our Dirac measures.
These combinations are not arbitrary; they have to be finite and they have to sum up to 1 (those are the convex combinations).
By working with the convex hulls, we can understand how various points and measures come together to form a more complex set.

Lebesgue Measure

Lebesgue Measure is a concept from measure theory, which is a way to assign a size or 'measure' to different sets, particularly in the context of integrations.
The Lebesgue measure, \(\lambda\), on \([0,1]\) assigns lengths to intervals in this space.
So, the integral of a function with respect to this measure gives a more accurate picture than simple sums, especially for continuous functions.
In this problem, \(\lambda\) represents a continuous measure, which is quite different from our discrete Dirac measures.
One key insight is that \(\lambda(f)\) can be close to 1, but no finite convex combination of Dirac measures can approximate this in such a way.

Radon Measures

Radon measures extend some of the ideas of Lebesgue measures but in a broader context.
They allow us to integrate even more general functions over more complex spaces.
In \(C[0,1]^*\), Radon measures help us link the continuous and discrete worlds by representing linear functionals.
For our unit ball \(B_{C[0,1]^{*}}\), these measures can either be Dirac measures (discrete) or something more elaborate like the Lebesgue measure \(\lambda\).
However, our exercise shows that even Radon measures have their limits, as \(\lambda\) cannot be represented within our convex hull of extremal points.
This insight provides a clearer boundary between different types of measures and spaces in functional analysis.