Question

If \(\nu\) is an arbitrary signed measure and \(\mu\) is a \(\sigma\)-finite measure on \((X, \mathcal{M})\) such that \(\nu \ll \mu\), there exists an extended \(\mu\)-integrable function \(f: X \rightarrow[-\infty, \infty]\) such that \(d \nu=f d \mu\). Hints: a. It suffices to assume that \(\mu\) is finite and \(\nu\) is positive. b. With these assumptions, there exists \(E \in \mathcal{M}\) that is \(\sigma\)-finite for \(\nu\) such that \(\mu(E) \geq \mu(F)\) for all sets \(F\) that are \(\sigma\)-finite for \(\nu\). c. The Radon-Nikodym theorem applies on \(E\). If \(F \cap E=\mathscr{\text { , then either }}\) \(\nu(F)=\mu(F)=0\) or \(\mu(F)>0\) and \(|\nu(F)|=\infty\).

Short answer

The function \(f\), derived from the Radon-Nikodym theorem, satisfies \(d\nu = f d\mu\).

Step-by-step solution

Understanding the Problem

We are given a signed measure \(u\) and a \(\sigma\)-finite measure \(\mu\), with the condition \(u \ll \mu\). Our task is to find an extended \(\mu\)-integrable function \(f\) such that \(du = f d\mu\). The problem suggests assuming \(\mu\) is finite and \(u\) is positive as a simplification.

Apply the Simplifications

To use the hints effectively, first assume that \(\mu\) is finite and that \(u\) is a positive measure. These assumptions allow us to simplify the application of the Radon-Nikodym theorem.

Identify \(E\) using \(\sigma\)-finiteness

Using the second hint, find \(E \in \mathcal{M}\) that is \(\sigma\)-finite for \(u\), ensuring that \(\mu(E) \geq \mu(F)\) for any set \(F\) that is \(\sigma\)-finite for \(u\). This set \(E\) plays a crucial role in applying the Radon-Nikodym theorem.

Apply Radon-Nikodym Theorem on \(E\)

Since \(u\) is positive and \(u \ll \mu\) within \(E\), apply the Radon-Nikodym theorem to find a function \(f: E \to [0, \infty]\) such that \(du = f d\mu\) on \(E\). This function \(f\) will be \(\mu\)-integrable because \(\mu(E)\) is finite.

Extend \(f\) Outside \(E\)

For any \(F ot\subset E\) (i.e., where \(F \cap E = \emptyset\)), determine based on the measure relation \(\mu(F) = 0\) or \(\mu(F) > 0\) that either \(u(F) = 0\) or \(u(F) = \infty\). Define \(f = 0\) on the set where \(\mu\) is zero.

Conclusion

The function \(f\) is now defined on the whole space \(X\) and satisfies the requirement \(du = f d\mu\). \(f\) is \(\mu\)-integrable since it is defined based on the Radon-Nikodym derivative on \(E\) and elsewhere as zero.

Key concepts

Signed Measure

A signed measure is an extension of the concept of a regular measure, allowing for the allocation of negative values in addition to the usual non-negative values. This extends the utility of measures in mathematical analysis. In a measure space, a signed measure \( u \) assigns a real number, which could be positive, negative, or zero, to a subset of a given set. Unlike regular measures, signed measures can take on negative values, offering a broader scope for mathematics, particularly in the realms of probability and financial modeling.
One key aspect of signed measures is how they relate to positive measures. If a signed measure is expressed as the difference of two finite measures, \( u = u^+ - u^- \), \( u^+ \) and \( u^- \) are positive measures indicating positive and negative contributions to \( u \) respectively. This decomposition is valuable in deeper mathematical analysis involving the Radon-Nikodym theorem, which helps in defining densities with respect to another measure.

Sigma-Finite Measure

A sigma-finite measure is crucial in measure theory, especially when dealing with infinite measure spaces. A measure \( \mu \) is termed \( \sigma \)-finite if the space can be divided into a countable union of sets with finite measure. This property simplifies managing infinite measures by breaking them into more manageable, finite parts.
This is especially useful in applying the Radon-Nikodym theorem, which often needs \( \mu \) to be \( \sigma \)-finite for the theorem to hold. Consider \( X \) as a space, it is \( \sigma \)-finite under measure \( \mu \) if \( X = \bigcup_{i=1}^{\infty} A_i \) with each \( \mu(A_i) < \infty \). This decomposition into finite parts allows for the application of analysis techniques that are feasible in finite scenarios, while still covering the entire space.

Measure Theory

Measure theory is a fundamental mathematical theory concerned with the assignment of a consistent measure to subsets of a given set. It forms the backbone of modern probability and integration, providing a framework for understanding and working with infinite processes.
In essence, measure theory generalizes the notion of 'length' or 'volume' to abstract spaces, allowing for rigorous treatment of concepts like area, length, or probability within a unified framework. This theory is crucial when dealing with integrals, especially in defining the integral of functions that are not necessarily continuous or defined everywhere. Concepts like \( \sigma \)-algebras, \( \sigma \)-finite measures, and signed measures all fall under this umbrella, offering tools for managing otherwise complex mathematical structures.

Integrable Function

An integrable function is a function that can be integrated over the space it is defined on, with respect to a given measure. Integrability signifies that the integral of the function, often interpreted as an area or volume under the function's curve, is finite. In simple terms, for a function \( f \) to be \( \mu \)-integrable, the integral \( \int |f| \, d\mu < \infty \). This is the key condition that allows us to perform meaningful analysis and calculations using this function across the space.
In the context of the Radon-Nikodym theorem, finding a \( \mu \)-integrable function is crucial to express one measure in terms of another more easily handled measure via their respective densities. This allows for the transformation of complicated measures into simpler forms through integrable functions.