ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Binary cartesian set product
▼ Binary relation
▼ Map
▼ Function
▼ Measure
▼ Real measure
▼ Euclidean real measure
▼ Complex measure
▼ Basic measure
▼ Unsigned basic measure
▼ Absolutely continuous measure
Definition D3567
Measure absolute continuity relation
Formulation 2
Let $M = (X, \mathcal{F})$ be a D1108: Measurable space such that
(i) $\mathcal{M} : = \mathcal{M}(M)$ is the D3566: Set of unsigned basic measures on $M$
The absolute continuity relation on $\mathcal{M}$ is the D4: Binary relation \begin{equation} {\ll} : = \left\{ (\mu, \nu) \in \mathcal{M} \times \mathcal{M} \mid \forall \, E \in \mathcal{F} \left( \mu(E) = 0 \quad \implies \quad \nu(E) = 0 \right) \right\} \end{equation}