Question

If \(\mu\) is a positive Radon measure on \(X\) with \(\mu(X)=\infty\), there exists \(f \in C_{0}(X)\) such that \(\int f d \mu=\infty\). Consequently, every positive linear functional on \(C_{0}(X)\) is bounded.

Short answer

Such a function exists and implies positive linear functionals on \( C_0(X) \) are bounded.

Step-by-step solution

Understanding the Exercise

We need to prove that for a positive Radon measure \( \mu \) on a space \( X \) with \( \mu(X) = \infty \), there exists a function \( f \in C_0(X) \) such that the integral \( \int f d\mu = \infty \). Secondly, this statement implies that every positive linear functional on \( C_0(X) \) is bounded.

Defining the Function

To find such a function \( f \), consider \( f_n = \chi_{E_n} \), where \( E_n \subset X \) are disjoint measurable sets with \( \mu(E_n) = \infty \). Choose \( E_n \) such that \( \mu(E_n) \rightarrow \infty \) as \( n \rightarrow \infty \).

Constructing the Function

Define the function \( f \) as \( f(x) = \sum_{n=1}^{\infty} \frac{1}{2^n} f_n(x) \). Then \( f \in C_0(X) \) because it is a finite linear combination of indicator functions belonging to \( C_0(X) \).

Evaluating the Integral

Consider \( \int f d\mu = \int \sum_{n=1}^{\infty} \frac{1}{2^n} f_n(x) d\mu \). By Tonelli's theorem and the disjointness of \( E_n \), this equals \( \sum_{n=1}^{\infty} \frac{1}{2^n} \int f_n d\mu = \sum_{n=1}^{\infty} \frac{1}{2^n} \mu(E_n) = \infty \).

Implication for Positive Linear Functionals

A positive linear functional \( L \) on \( C_0(X) \) is bounded because if it were not, for some \( f \,\geq\, 0 \), \( L(f) = \infty \). This contradicts the assumption that \( L \) is positive and linear on \( C_0(X) \), enforcing it to be bounded by \( \mu \).

Key concepts

Positive Linear Functional

A positive linear functional is a special kind of operation applied to a space of functions. Imagine it as a rule that assigns a real number to each function, but unlike any random rule, it has some specific properties:

• Positivity: If you give it a non-negative function, it will return a non-negative number. This makes sense because you wouldn't expect to get a negative outcome from something that starts out non-negative!
• Linearity: This means that if you take two functions and the rule processes them together, the result is the same as processing them separately and then combining the results. Mathematically, for functions \(f\) and \(g\) and scalars \(a\) and \(b\), if \(L\) is a linear functional, then \(L(af + bg) = aL(f) + bL(g)\).

In the context of this exercise, any positive linear functional you examine is not just positive and linear but also bounded, which simplifies many calculations you perform on functions in this setting.

Integral of Functions

The integral of a function is a fundamental building block in calculus and analysis. It signifies the total area under the curve that the function creates on a graph. This total can be finite or infinite based on the behavior of the function and the space it describes.

• When discussing integrals, especially with respect to measures like the Radon measure, the result gives a sense of the 'overall impact' or 'size' of the function over its domain.
• In the case presented in the exercise, the function \(f\) maps to an infinite value when integrated, highlighting the concept's role in determining when a function or measure is infinite.

Understanding integration helps you map out how functions are summing up over their space, lending insight into not just their "footprint" but also their potential behavior in that space.

Bounded Functional

A bounded functional is an important property in functional analysis related to the control and predictability of linear functionals. To say a functional is bounded implies that there is a cap—a maximum value—on how large the results of the functional can be, across all inputs of a given size.

• This means that no matter what function you apply to it, the output value is limited and won't shoot off to infinity, provided the input function is properly scaled.
• Mathematically, a functional \(L\) is bounded if there exists some constant \(C\) such that for all functions \(f\), \(|L(f)| \leq C\|f\|\), where \(\|f\|\) is some measure of the size of \(f\).
• In this exercise, the importance lies in showing how positive linear functionals, when combined with continuity and compact support, remain bounded, ensuring predictable and consistent computation behaviors.

A bounded functional helps in maintaining a predictable scope of results which simplifies analytical and computational processes.

Continuous Functions with Compact Support

Continuous functions with compact support, denoted as \(C_0(X)\), play a critical role in analysis, providing a collection of functions that are both controlled and finite.

• Continuous functions: These functions have no breaks, jumps, or irregularities, allowing for smooth transitions from one point to another.
• Compact support: This means that even though the entire function might extend over an infinite domain, it actually "vanishes" to zero outside a finite region. Thus, the function does not affect regions far outside a central area.
• In practical terms, these kinds of functions are easier to work with because they disappear at infinity, curbing potential complications that arise from infinite domains or wild behavior at boundaries.

Within this exercise, use of such functions ensures any manipulations or operations on the function itself do not extend beyond a finite context, keeping everything within manageable limits.