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The support of a singular measure is the smallest closed set for which the measure of its complement is zero. The support is often a set of Lebesgue measure zero.

Two measures $\mu$ and $\nu$ are mutually singular, denoted by $\mu \perp \nu$, if there exist two disjoint measurable sets $A$ and $B$ such that $\mu$ is zero on $B$ and $\nu$ is zero on $A$.

The Cantor measure is a classic example of a singular measure. It is defined on the Cantor set, which has Lebesgue measure zero, yet the Cantor measure assigns it a measure of one.

The Radon-Nikodym theorem states that if a measure $\nu$ is absolutely continuous with respect to measure $\mu$, there exists a Radon-Nikodym derivative. Singular measures do not have a Radon-Nikodym derivative with respect to Lebesgue measure.

A singular measure is a type of measure that is orthogonal to absolutely continuous measures with respect to Lebesgue measure. It assigns measure zero to sets of Lebesgue measure zero.

Singular continuous functions are functions whose derivative exists almost everywhere and is zero almost everywhere. The distribution function of a singular measure is a singular continuous function.

The Lebesgue decomposition theorem states that any measure $\nu$ can be uniquely written as the sum of two measures $\nu = \nu_{ac} + \nu_{s}$, where $\nu_{ac}$ is absolutely continuous with respect to Lebesgue measure and $\nu_{s}$ is singular with respect to Lebesgue measure.