[Math] How is a singular continuous measure defined

measure-theoryreal-analysisreference-requestsingular-measures

On a measurable space, how is a measure being singular continuous relative to another defined? I searched on the internet and in some books to no avail and it mostly appears in a special case – the Lebesgue measure space $\mathbb{R}$.

  1. Do you know if singular continuous measures can be generalized to a
    more general measure space than Lebesgue measure space $\mathbb{R}$?
    In particular, can it be defined on any measure space, as hinted by
    the Wiki article I linked below?
  2. The purpose of knowing the answers to previous questions is that I would like to know to
    what extent the decomposition of a singular measure into a discrete
    measure and a singular continuous measure still exist, all wrt a refrence measure?

Thanks and regards!

PS: In case you may wonder, I encounter this concept from Wikipedia (feel it somehow sloppy though):

Given $μ$ and $ν$ two σ-finite signed measures on a measurable space
$(Ω,Σ)$, there exist two $σ$-finite signed measures $ν_0$ and $ν_1$
such that:

  • $\nu=\nu_0+\nu_1\,$
  • $\nu_0\ll\mu$ (that is, $ν_0$ is absolutely continuous with respect to $μ$)
  • $\nu_1\perp\mu$ (that is, $ν_1$ and $μ$ are singular).

The decomposition of the singular part can refined: $$
\, \nu = \nu_{\mathrm{cont}} + \nu_{\mathrm{sing}} + \nu_{\mathrm{pp}} $$ where

  • $\nu_{\mathrm{cont}}$ is the absolutely continuous part
  • $\nu_{\mathrm{sing}}$ is the singular continuous part
  • $\nu_{\mathrm{pp}}$ is the pure point part (a discrete measure).

Best Answer

I am not completely sure, and I cannot provide a publicly available reference, but I read in some lecture notes from our university that this decomposition can be generalized to $\mathbb{R}^n$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^n$ and $\mu$ the measure under consideration. Then,

$$ \mu = \mu_a + \mu_s + \mu_d $$

where $\mu_d$ is discrete (i.e., supported on a countable set, with positive measure for every atom), $\mu_a$ is absolutely continuous w.r.t. $\lambda$ (i.e., it possesses a density), and $\mu_s$ is singularly continuous, i.e., it is supported on a Lebesgue null-set, and the atoms of this set have zero measure.

An example for $\mu_s$ in $\mathbb{R}^2$ would be a measure which is supported on a one-dimensional submanifold of $\mathbb{R}^2$, e.g., the uniform distribution on the unit circle.

Related Question