Mixing
Is it true that $mathcal H^{n-1}(partial (A cap B))=mathcal H^{n-1}((partial A) cap B) + mathcal H^{n-1}(...
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Let $A$ and $B$ subsets of $mathbb R ^n$, $B=B(x,r)$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $mathbb R ^n$ by $mathcal H^{n-1}$. Also assume that $mathcal H^{n-1} (partial A) < + infty$ (One can assume that $A$ is a set of finite perimeter in necessary).
In this case, is it the following identity holds?
$ mathcal H^{n-1}(partial (A cap B))= mathcal H^{n-1}((partial A) cap B) + mathcal H^{n-1}( Acap ( partial B))$
real-analysis general-topology measure-theory geometric-measure-theory
asked Jan 23 at 9:56
HumedHumed
797
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add a comment |
$begingroup$
Let $A$ and $B$ subsets of $mathbb R ^n$, $B=B(x,r)$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $mathbb R ^n$ by $mathcal H^{n-1}$. Also assume that $mathcal H^{n-1} (partial A) < + infty$ (One can assume that $A$ is a set of finite perimeter in necessary).
In this case, is it the following identity holds?
$ mathcal H^{n-1}(partial (A cap B))= mathcal H^{n-1}((partial A) cap B) + mathcal H^{n-1}( Acap ( partial B))$
real-analysis general-topology measure-theory geometric-measure-theory
asked Jan 23 at 9:56
HumedHumed
797
$endgroup$
$begingroup$
What if $A=Bbb R^nsetminus B$?
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– Alex Ravsky
Jan 26 at 23:43
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@AlexRavsky Good point! in this case only the second term on the right is non-zero. Let's see the answer in mathoverflow.net/questions/321524/… . In Brief, one should replace the first Hausdorff measure with perimeters, and then the result is holds for almost every balls. $P(A cap B_{rho}) = P(A, B_{rho}) + mathcal H^{n-1}(A cap partial B_{rho})$, a.e. $rho$. (see the full answer in the link above)
$endgroup$
– Humed
Jan 28 at 6:56
add a comment |
$begingroup$
Let $A$ and $B$ subsets of $mathbb R ^n$, $B=B(x,r)$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $mathbb R ^n$ by $mathcal H^{n-1}$. Also assume that $mathcal H^{n-1} (partial A) < + infty$ (One can assume that $A$ is a set of finite perimeter in necessary).
In this case, is it the following identity holds?
$ mathcal H^{n-1}(partial (A cap B))= mathcal H^{n-1}((partial A) cap B) + mathcal H^{n-1}( Acap ( partial B))$
real-analysis general-topology measure-theory geometric-measure-theory
asked Jan 23 at 9:56
HumedHumed
797
$endgroup$
Let $A$ and $B$ subsets of $mathbb R ^n$, $B=B(x,r)$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $mathbb R ^n$ by $mathcal H^{n-1}$. Also assume that $mathcal H^{n-1} (partial A) < + infty$ (One can assume that $A$ is a set of finite perimeter in necessary).
In this case, is it the following identity holds?
$ mathcal H^{n-1}(partial (A cap B))= mathcal H^{n-1}((partial A) cap B) + mathcal H^{n-1}( Acap ( partial B))$
real-analysis general-topology measure-theory geometric-measure-theory
real-analysis general-topology measure-theory geometric-measure-theory
asked Jan 23 at 9:56
HumedHumed
797
asked Jan 23 at 9:56
HumedHumed
797
edited Jan 23 at 10:54
Humed
asked Jan 23 at 9:56
HumedHumed
797
asked Jan 23 at 9:56
HumedHumed
797
asked Jan 23 at 9:56
HumedHumed
797
797
$begingroup$
What if $A=Bbb R^nsetminus B$?
$endgroup$
– Alex Ravsky
Jan 26 at 23:43
$begingroup$
@AlexRavsky Good point! in this case only the second term on the right is non-zero. Let's see the answer in mathoverflow.net/questions/321524/… . In Brief, one should replace the first Hausdorff measure with perimeters, and then the result is holds for almost every balls. $P(A cap B_{rho}) = P(A, B_{rho}) + mathcal H^{n-1}(A cap partial B_{rho})$, a.e. $rho$. (see the full answer in the link above)
$endgroup$
– Humed
Jan 28 at 6:56
add a comment |
$begingroup$
What if $A=Bbb R^nsetminus B$?
$endgroup$
– Alex Ravsky
Jan 26 at 23:43
$begingroup$
@AlexRavsky Good point! in this case only the second term on the right is non-zero. Let's see the answer in mathoverflow.net/questions/321524/… . In Brief, one should replace the first Hausdorff measure with perimeters, and then the result is holds for almost every balls. $P(A cap B_{rho}) = P(A, B_{rho}) + mathcal H^{n-1}(A cap partial B_{rho})$, a.e. $rho$. (see the full answer in the link above)
$endgroup$
– Humed
Jan 28 at 6:56
$begingroup$
What if $A=Bbb R^nsetminus B$?
$endgroup$
– Alex Ravsky
Jan 26 at 23:43
$begingroup$
What if $A=Bbb R^nsetminus B$?
$endgroup$
– Alex Ravsky
Jan 26 at 23:43
$begingroup$
@AlexRavsky Good point! in this case only the second term on the right is non-zero. Let's see the answer in mathoverflow.net/questions/321524/… . In Brief, one should replace the first Hausdorff measure with perimeters, and then the result is holds for almost every balls. $P(A cap B_{rho}) = P(A, B_{rho}) + mathcal H^{n-1}(A cap partial B_{rho})$, a.e. $rho$. (see the full answer in the link above)
$endgroup$
– Humed
Jan 28 at 6:56
$begingroup$
@AlexRavsky Good point! in this case only the second term on the right is non-zero. Let's see the answer in mathoverflow.net/questions/321524/… . In Brief, one should replace the first Hausdorff measure with perimeters, and then the result is holds for almost every balls. $P(A cap B_{rho}) = P(A, B_{rho}) + mathcal H^{n-1}(A cap partial B_{rho})$, a.e. $rho$. (see the full answer in the link above)
$endgroup$
– Humed
Jan 28 at 6:56
add a comment |
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$begingroup$
What if $A=Bbb R^nsetminus B$?
$endgroup$
– Alex Ravsky
Jan 26 at 23:43
$begingroup$
@AlexRavsky Good point! in this case only the second term on the right is non-zero. Let's see the answer in mathoverflow.net/questions/321524/… . In Brief, one should replace the first Hausdorff measure with perimeters, and then the result is holds for almost every balls. $P(A cap B_{rho}) = P(A, B_{rho}) + mathcal H^{n-1}(A cap partial B_{rho})$, a.e. $rho$. (see the full answer in the link above)
$endgroup$
– Humed
Jan 28 at 6:56