From the definition I am using and restrict to $\mathbb{R}^d$ only, what can we say about $\sigma$-algebra of $\nu$-measurable sets, $\mathfrak{B}_{\nu}$?

Some more specefic questions:

  1. It contains all Borel sets. If $\mu$ and $\mathfrak{B}_{\mu}$ denote Lebesgue and $\sigma$-algebra of Lebesgue-measurable sets respectively, what is the relation between $\mathfrak{B}_{\mu}$ and $\mathfrak{B}_{\nu}$?
  2. (meta question) Is it interesting to study $\mathfrak{B}_{\nu}$? The book I am reading seems to delve into "density" of measure $\nu(E)$ w.r.t $\mu(E)$.

As far as 1. is concerned it is not clear how you construct $\mathfrak{B}_{\mu}$ and $\mathfrak{B}_{\nu}$. The Borel-$\sigma$-field does not depend on the measure but on the topology, which defines open sets and those induce the Borel-$\sigma$-field. As far as my experience goes you usually define measures on $\sigma$-fields, not conversely. Thus you are comparing identical objects, unless you explain what topology you are using to get $\mathfrak{B}_{\nu}$ (I assume that for $\mathfrak{B}_{\mu}$ you are using a $d$-norm to define a metric and this metric defines open sets on $\mathbb{R}^d$).

I cannot comment on your question, that's why I'm posting an answer.

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  • $\begingroup$ $\mathfrak{B}_{\nu}$ is constructed using Caratheodory extension. $\endgroup$ – Lei Lei Apr 7 '13 at 4:03

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