TITLE: How is a singular continuous measure defined?
QUESTION [15 upvotes]: 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 $\mathbb{R}$. 

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?
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).

REPLY [2 votes]: A singular (say, probability) measures $\mu$ with respect to the Lebesgue measure $\lambda$ on $\mathbb R^d$ satisfies by definition: there exists a Borel set $S$ such that $\mu(S)=1$ and $\lambda(S)=0$. To obtain a continuous singular measure, that is satisfying $\mu(\{x\})=0$ for any $x\in\mathbb R^d$, the idea is to find a measure supported on a set $S$ having positive dimension but strictly less than $d$. When $d=1$, you can use sets of fractional dimension, like the cantor set, but in higher dimension you can find more easy examples, e.g. taking for $\mu$ a measure on the unit circle (resp. sphere) in $\mathbb R^2$ (resp. $\mathbb R^d$).