Definition:Absolute Continuity

Definition

Real Function

Let $I \subseteq \R$ be a real interval.

A real function $f: I \to \R$ is said to be absolutely continuous if and only if it satisfies the following property:

For every $\epsilon > 0$ there exists $\delta > 0$ such that the following property holds:

For every finite set of pairwise disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ such that:

$\ds \sum_{i \mathop = 1}^n \size {b_i - a_i} < \delta$

it holds that:

$\ds \sum_{i \mathop = 1}^n \size {\map f {b_i} - \map f {a_i} } < \epsilon$

Measure

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ and $\nu$ be measures on $\struct {X, \Sigma}$.

We say that $\nu$ is absolutely continuous with respect to $\mu$ and write:

$\nu \ll \mu$

if and only if:

$\ds \forall A \in \Sigma : \map \mu A = 0 \implies \map \nu A = 0$

Signed Measure

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a measure on $\struct {X, \Sigma}$.

Let $\nu$ be a signed measure on $\struct {X, \Sigma}$.

Let $\size \nu$ be the variation of $\nu$.

We say that $\nu$ is absolutely continuous with respect to $\mu$ if and only if:

$\size \nu$ is absolutely continuous with respect to $\mu$.

We write:

$\nu \ll \mu$

Complex Measure

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a measure on $\struct {X, \Sigma}$.

Let $\nu$ be a complex measure on $\struct {X, \Sigma}$.

Let $\size \nu$ be the variation of $\nu$.

We say that $\nu$ is absolutely continuous with respect to $\mu$ if and only if:

$\size \nu$ is absolutely continuous with respect to $\mu$.

We write:

$\nu \ll \mu$