Lebesgue Decomposition Theorem

• categories: Measure Theory, Theorem

The Lebesgue Decomposition Theorem is a result in measure theory that provides a way to decompose any signed measure with respect to another measure into two mutually exclusive parts: one that is absolutely continuous with respect to the second measure and one that is singular with respect to it. This theorem is fundamental for understanding how one measure relates to another, especially when dealing with signed or complex measures.

Statement

Let $(X,\Sigma )$ be a measurable space, and let $\nu$ be a $\sigma$-finite positive measure on $(X,\Sigma )$. Then, for any $\sigma$-finite signed measure $\mu$ on $(X,\Sigma )$, there exist two unique signed measures ${\mu }_{ac}$ and ${\mu }_s$ such that:

1. $\mu ={\mu }_{ac}+{\mu }_s$,
2. ${\mu }_{ac}$ is absolutely continuous with respect to $\nu$ (written ${\mu }_{ac}\ll \nu$),
3. ${\mu }_s$ is singular with respect to $\nu$ (written ${\mu }_s\perp \nu$).

This decomposition $\mu ={\mu }_{ac}+{\mu }_s$ is called the Lebesgue decomposition of $\mu$ with respect to $\nu$.

Definitions

1. Absolutely Continuous Measure: A measure $\mu$ is absolutely continuous with respect to $\nu$ (denoted $\mu \ll \nu$) if, for every measurable set $A$, $\nu (A)=0$ implies $\mu (A)=0$.
2. Singular Measure: A measure $\mu$ is singular with respect to $\nu$ (denoted $\mu \perp \nu$) if there exists a measurable set $S$ such that $\mu$ is concentrated on $S$ and $\nu$ is concentrated on $S^c$, i.e., $\mu (S^c)=0$ and $\nu (S)=0$.

Intuition

The Lebesgue Decomposition Theorem splits the measure $\mu$ into two parts relative to $\nu$:

• The absolutely continuous part ${\mu }_{ac}$ corresponds to the component of $\mu$ that “follows” $\nu$ in the sense that if $\nu$ has no “weight” on a set, then neither does ${\mu }_{ac}$.
• The singular part ${\mu }_s$ corresponds to the component of $\mu$ that is entirely “orthogonal” to $\nu$, meaning ${\mu }_s$ is concentrated on a set that $\nu$ “ignores” (i.e., a set of $\nu$-measure zero).

Example

Suppose $\nu$ is the Lebesgue measure on $\mathbb{R}$, and let $\mu$ be a measure that assigns length (Lebesgue measure) to intervals plus an additional Dirac measure at a point $x_0\in \mathbb{R}$. Then:

• The absolutely continuous part ${\mu }_{ac}$ would be the restriction of $\mu$ to intervals (i.e., Lebesgue measure),
• The singular part ${\mu }_s$ would be the Dirac measure at $x_0$, which is concentrated at a single point and therefore does not overlap with the Lebesgue measure.

Applications

1. Radon-Nikodym Theorem: The absolutely continuous part ${\mu }_{ac}$ is significant in the Radon-Nikodym Theorem, which states that there exists a density function $f\in L^1(\nu )$ such that ${\mu }_{ac}(A)={\int }_{A}fd\nu$ for all measurable $A$.
2. Probability Theory: The decomposition helps in distinguishing between continuous and discrete parts of probability measures.
3. Signal Processing and Statistics: The theorem aids in separating noise (often singular) from a signal (typically absolutely continuous) with respect to a reference measure.

Proof Outline

1. Hahn Decomposition: Use the Hahn Decomposition Theorem to create sets where $\mu$ is positive and negative relative to $\nu$.
2. Define Components: Define ${\mu }_{ac}$ by using the Radon-Nikodym derivative with respect to $\nu$ (ensuring ${\mu }_{ac}\ll \nu$) and let ${\mu }_s=\mu -{\mu }_{ac}$.
3. Uniqueness: The uniqueness of ${\mu }_{ac}$ and ${\mu }_s$ follows from the properties of absolute continuity and singularity, as they describe mutually exclusive behaviors with respect to $\nu$.