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Given a borelian measure in $\mathbb{R}^2$, there is a canonical way or simply a way to obtain a measure on a line, for example $x=0$? (a measure with support in the line I'm considering). The question is very general, I explain this with this example because is clearly easy to understand.

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    Hausdorff measure is a possible (a bit overkill) approach2012-08-09

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Given $\mu$ a Borel measure on $\Bbb R^2$, define $\nu:{\cal B}_{\Bbb R}\to [0,\infty]$ by $\nu(E)=\mu(E\times [0,1]).$ Then $\nu$ is a Borel measure on $\Bbb R$.