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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).
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    It is a biiiigg mess-up. In fact you first decompose w.r.t. another measure then after having done that you can decompose into continuous and discrete or atomic and atomless or anything in between. It happens and that is where probably confusion comes from is that the Lebesgue measure itself is already purely continuous atomless and so on so the decomposition of absolutely continuous part will stay the same you might also call it somewhat regular continuous and regular discrete as well as singular continuous and sincular discrete. See the analogues?2014-11-06

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