Singular measure

A singular measure is a term from the mathematical branch of measure theory. It plays a major role in the classification of measures regarding another dimension and finds particular application to the decomposition theorem of Lebesgue and the representation theorem in the stochastic.

Definition

A measure is called singular with respect to another dimension (even singular to or -singular ) if there are a lot of

Here are the dimensions and defined on the same measurement space. For " is singular with respect to " write short one.

Examples

• The zero degree with respect to each other on any measure measurement space singular.
• Each Dirac measure on with respect to the Lebesgue measure singular.
• Each discrete distribution with respect to the Lebesgue measure singular.
• The Cantor distribution on the measurement space is a continuous, singular distribution with respect to the Lebesgue measure.

Properties

The singularity of moderation is a symmetric relation. It is therefore