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Let $A$ be a (bounded) Borel set in $R^n$. Then we know that its projection $A_1$ on $R^{n-1}$ does not have to be Borel. But does $A_1$ have the following property?

Let $\mu$ be a given nonnegative, finite Borel measure on $R^{n-1}$. Then:

$\forall\epsilon>0 \ \exists B,C\subset R^{n-1}$ Borel sets, such that $B\subset A_1\subset C$ and $\mu(C\setminus B)<\epsilon$.

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Yes. Projections of Borel sets are analytic sets, and these are measurable with respect to completions of Borel measures. (In fact, it will be possible to get $\mu(C\setminus B)=0$.)

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  • $\begingroup$ Looks like Bob may find a book on descriptive set theory to be of interest. $\endgroup$ Oct 11, 2014 at 20:10
  • $\begingroup$ Indeed, could you recommend a book to look this up? User friendly to PDE people, please ... Thanks! $\endgroup$
    – Bob
    Oct 11, 2014 at 20:30
  • $\begingroup$ @Bob, a good book with what you need at the moment is "A second course on real functions", by van Rooij and Schikhof. This is an analysis book, and basic descriptive set theory is covered "classically", in the context of $\mathbb R$, as opposed to arbitrary Polish spaces. If you find yourself needing more than what is there, the best suggestion is Kechris's book "Classical descriptive set theory". $\endgroup$ Oct 12, 2014 at 1:41

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