Absolute continuity

• categories: Measure Theory, Definition

Absolute Continuity

In measure theory, absolute continuity describes a specific relationship between two measures. Intuitively, a measure $\mu$ is absolutely continuous with respect to another measure $\nu$ if $\mu$ “follows” $\nu$ in the sense that $\mu$ assigns no “weight” to any set where $\nu$ has zero measure.

Definition

Let $(X,\Sigma )$ be a measurable space, and let $\mu$ and $\nu$ be two measures on $(X,\Sigma )$. The measure $\mu$ is said to be absolutely continuous with respect to $\nu$ (written $\mu \ll \nu$) if:

$$\text{for any measurable set }A\subset X,\nu (A)=0\Rightarrow \mu (A)=0.$$

In other words, $\mu$ “inherits” the null sets of $\nu$.

Examples

1. Lebesgue Measure and Weighted Measures: Let $\nu$ be the Lebesgue measure on $\mathbb{R}$, and let $\mu$ be defined by $\mu (A)={\int }_{A}f(x)dx$ for a non-negative function $f\in L^1(\mathbb{R})$. Then $\mu \ll \nu$, because any set of Lebesgue measure zero will also have zero measure under $\mu$.

2. Discrete vs. Continuous Measures: If $\nu$ is the Lebesgue measure and $\mu$ is the Dirac delta measure at some point $x_0\in \mathbb{R}$ (defined by $\mu (A)=1$ if $x_0\in A$ and $\mu (A)=0$ otherwise), then $\mu$ is not absolutely continuous with respect to $\nu$, as $\mu$ assigns weight to a set (the single point $\{x_0\}$) that has Lebesgue measure zero.

Properties

1. Preservation of Null Sets: Absolute continuity implies that $\mu$ assigns zero measure to every $\nu$-null set, but not necessarily vice versa.
2. Total Variation: If $\mu$ is absolutely continuous with respect to $\nu$, then the total variation of $\mu$ is also absolutely continuous with respect to $\nu$.
3. Implication for Functions: In the context of real analysis, if a function $F$ is absolutely continuous on an interval $[a,b]$, then it has a derivative almost everywhere, and $F$ can be represented as an integral of its derivative.