Singular measures

• categories: Measure Theory, Definition

Singular Measures

• Definition: A measure $\mu$ on a measurable space $(X,\mathcal{A})$ is said to be singular with respect to another measure $\nu$ (denoted $\mu \perp \nu$) if there exists a set $A\in \mathcal{A}$ such that:

  • $\mu (A^c)=0$ (i.e., $\mu$ is concentrated on $A$)
  • $\nu (A)=0$ (i.e., $\nu$ is concentrated on $A^c$)

• Intuition: Singular measures do not share common support; one measure assigns positive measure to a set where the other measure assigns zero and vice versa.

• Example:

  • The Lebesgue measure $\lambda$ on $\mathbb{R}$ and the Dirac measure $\delta$ at a point $x_0\in \mathbb{R}$ are singular, as $\lambda (\{x_0\})=0$ and $\delta (\mathbb{R}\setminus \{x_0\})=0$.

• Properties:

  • If $\mu \perp \nu$, then there exists a partition $X=A\cup A^c$ such that $\mu$ is supported on $A$ and $\nu$ on $A^c$.
  • Singular measures often arise in contexts where continuous and discrete measures coexist.