▾ 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
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}