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 it satisfies the following property:

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

For every finite set of 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 $M$ be a measurable space.

Let $\mu_1$ and $\mu_2$ be measures on $M$.

Let $\map {\mu_1} E = 0$ whenever $\map {\mu_2} E = 0$.

Then $\mu_1$ is absolutely continuous with respect to $\mu_2$.