Hahn-Hellinger Theorem in Measure Theory

The Hahn-Hellinger theorem is a fundamental theorem in measure theory which complements knowledge of LebesgueIntegration; it is a result which provides a foundational understanding of the relationship between two measures in the context of their absolute continuity.

Statement of the Theorem

Given a measurable space $(\Omega, \Sigma)$, let $\mu$ and $\nu$ be two finite measures on $\Sigma$. This relationship, denoted $\mu \ll \nu$, means that for every $E \in \Sigma$ with $\nu(E) = 0$, we have $\mu(E) = 0 $.

The Hahn-Hellinger theorem can be articulated as:

If for every non-negative $\Sigma$-measurable function $f$ with $\int f d\nu = 0$ it holds that $\int f d\mu = 0$, then $\mu$ is absolutely continuous with respect to $\nu$.

This can equivalently be stated as:

If for every non-negative $\Sigma$-measurable function $f$ with $\int f d\nu = 0$, it holds that $\int f d\mu = 0$, then for every measurable set $E \in \Sigma$, $\mu(E) = 0$ whenever $\nu(E) = 0$.

Proof and Intuition

The theorem essentially articulates that if two measures concur on sets of measure zero, then they must also be in agreement on sets with positive measure.

The proof revolves around defining a non-negative indicator function for any set $E \in \Sigma$ such that $\nu(E) = 0$. The goal is to demonstrate that $\mu(E) = 0$ using properties of the Lebesgue integral.

Connection to the Radon-Nikodym Theorem

The Hahn-Hellinger theorem and the Radon-Nikodym theorem are closely knit. While the former characterizes absolute continuity, the latter provides an existence proof for a density function that expresses one measure in terms of another when one measure is absolutely continuous with respect to the other. The existence of a density function is significant because it allows us to express one measure in terms of the other, providing a deeper understanding of the relationship between the two measures.

Applications

The implications of the Hahn-Hellinger theorem span the field measure theory(and thus probability theory) and it is instrumental in the proof of the Radon-Nikodym theorem, among other pivotal results.

History

The theorem was first elucidated by Hans Hahn and Ernst Hellinger in the year 1910, marking a significant milestone in the realm of measure theory.