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.
How to "project" a measure
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measure-theory
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0Well, you could put, if $\pi$ is a projection on the line, $A$ a subset of the line $\bar{\mu}(A)=\mu(\pi^{-1}(A))$ where $\mu$ is your original measure (this suggests the natural $\sigma$-algebra to use). Another way could be to restrict $\mu$ to the line. – 2012-08-09
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2Jose, both approaches are bad. The first allways give infinite values for $\overline{\mu}(A)$. The second is meaningless. How do you restrict arbitrary 17-gon to a line? – 2012-08-09
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0@Norbert: The first not always, since we're only being told the measure is Borel (it could be finite). For the second, just put $\nu(A)=\mu(A\cap L)$ ($L$ being the line), this is what I meant by restriction (it gives a measure supported in the line). I agree though that they may not be the best, but they're the most natural, at least coming from a course in topology. – 2012-08-09
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0I had thought this two options. In the first one the problem is that in general you can not assume that the measure is finite, in the second case you don't get anything if the measures aren't supported on this line. – 2012-08-09
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0There is not a measure that you can define using a kind of limit to get close to the subset in the line you are interested in? – 2012-08-09
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0Hausdorff measure is a possible (a bit overkill) approach – 2012-08-09