Mixing
Measure of Subspace Swept Out by Another
Sorry, it's a difficult question to give a title to, so if someone can think of a better one then feel free. I am trying to figure out how to set up a problem.
Suppose we have two closed subsets $X, Y$ of a fixed $mathbb{R}^n$ and we fix a point $x_0 in X$, and consider a closed arc $A$ in $mathbb{R}^n setminus Y$ with left end point $x_0$ and right end point $x_1$ (the latter not necessarily in $X$). We may assume that $A$ is a smooth, or even analytic, embedding if it's helpful.
For each $a in A$ let $f_a$ be the affine shift of $mathbb{R}^n$ which sends $x_0$ to $a$. Thus we obtain a continuous family of maps on $X$ indexed by $[0,1]$ which will sweep out an area, when we restrict these to $X$.
This is what I want to calculate the measure of: $$lbrace x in X | f_a(x) notin Y text{ for any } a in A rbrace$$
In other words, I want to calculate the measure of the part of $X$ which never hits $Y$ as the space $X$ is dragged along the curve $A$. It is fine to assume that $mu(X), mu(Y) < infty$ and thus calculate its complement, instead. How the heck do I set this up? Is there an averaging method (a la Bogolyubov-Krillov) we can use? It is also fine to assume that $X$ and $Y$ are embedded $n$-manifolds with boundary, so in particular you could alternatively assume that $X, Y$ are open, or regular closed, etc. In other words, they're nice. Locally connected, simply connected, pretty much anything is fine.
Thanks a lot!
real-analysis euclidean-geometry lebesgue-integral geometric-measure-theory
edited Nov 21 '18 at 10:14
John B
12.2k51840
asked Nov 20 '18 at 23:02
John Samples
1,086516
add a comment |
Sorry, it's a difficult question to give a title to, so if someone can think of a better one then feel free. I am trying to figure out how to set up a problem.
Suppose we have two closed subsets $X, Y$ of a fixed $mathbb{R}^n$ and we fix a point $x_0 in X$, and consider a closed arc $A$ in $mathbb{R}^n setminus Y$ with left end point $x_0$ and right end point $x_1$ (the latter not necessarily in $X$). We may assume that $A$ is a smooth, or even analytic, embedding if it's helpful.
For each $a in A$ let $f_a$ be the affine shift of $mathbb{R}^n$ which sends $x_0$ to $a$. Thus we obtain a continuous family of maps on $X$ indexed by $[0,1]$ which will sweep out an area, when we restrict these to $X$.
This is what I want to calculate the measure of: $$lbrace x in X | f_a(x) notin Y text{ for any } a in A rbrace$$
In other words, I want to calculate the measure of the part of $X$ which never hits $Y$ as the space $X$ is dragged along the curve $A$. It is fine to assume that $mu(X), mu(Y) < infty$ and thus calculate its complement, instead. How the heck do I set this up? Is there an averaging method (a la Bogolyubov-Krillov) we can use? It is also fine to assume that $X$ and $Y$ are embedded $n$-manifolds with boundary, so in particular you could alternatively assume that $X, Y$ are open, or regular closed, etc. In other words, they're nice. Locally connected, simply connected, pretty much anything is fine.
Thanks a lot!
real-analysis euclidean-geometry lebesgue-integral geometric-measure-theory
edited Nov 21 '18 at 10:14
John B
12.2k51840
asked Nov 20 '18 at 23:02
John Samples
1,086516
Do we know the measure of $X$ and $Y$?
– Santana Afton
Nov 20 '18 at 23:43
Just some finite number, all measures are real. In the set-up for the application I have in mind, you would know the measure of $X$ and $Y$ at the outset; no inverse problem. I am looking for a method (probably some integral equation) that works for two fixed $X, Y$, doesn't have to be germane to every pair of subspaces simultaneously.
– John Samples
Nov 21 '18 at 1:04
add a comment |
Sorry, it's a difficult question to give a title to, so if someone can think of a better one then feel free. I am trying to figure out how to set up a problem.
Suppose we have two closed subsets $X, Y$ of a fixed $mathbb{R}^n$ and we fix a point $x_0 in X$, and consider a closed arc $A$ in $mathbb{R}^n setminus Y$ with left end point $x_0$ and right end point $x_1$ (the latter not necessarily in $X$). We may assume that $A$ is a smooth, or even analytic, embedding if it's helpful.
For each $a in A$ let $f_a$ be the affine shift of $mathbb{R}^n$ which sends $x_0$ to $a$. Thus we obtain a continuous family of maps on $X$ indexed by $[0,1]$ which will sweep out an area, when we restrict these to $X$.
This is what I want to calculate the measure of: $$lbrace x in X | f_a(x) notin Y text{ for any } a in A rbrace$$
In other words, I want to calculate the measure of the part of $X$ which never hits $Y$ as the space $X$ is dragged along the curve $A$. It is fine to assume that $mu(X), mu(Y) < infty$ and thus calculate its complement, instead. How the heck do I set this up? Is there an averaging method (a la Bogolyubov-Krillov) we can use? It is also fine to assume that $X$ and $Y$ are embedded $n$-manifolds with boundary, so in particular you could alternatively assume that $X, Y$ are open, or regular closed, etc. In other words, they're nice. Locally connected, simply connected, pretty much anything is fine.
Thanks a lot!
real-analysis euclidean-geometry lebesgue-integral geometric-measure-theory
edited Nov 21 '18 at 10:14
John B
12.2k51840
asked Nov 20 '18 at 23:02
John Samples
1,086516
Sorry, it's a difficult question to give a title to, so if someone can think of a better one then feel free. I am trying to figure out how to set up a problem.
Suppose we have two closed subsets $X, Y$ of a fixed $mathbb{R}^n$ and we fix a point $x_0 in X$, and consider a closed arc $A$ in $mathbb{R}^n setminus Y$ with left end point $x_0$ and right end point $x_1$ (the latter not necessarily in $X$). We may assume that $A$ is a smooth, or even analytic, embedding if it's helpful.
For each $a in A$ let $f_a$ be the affine shift of $mathbb{R}^n$ which sends $x_0$ to $a$. Thus we obtain a continuous family of maps on $X$ indexed by $[0,1]$ which will sweep out an area, when we restrict these to $X$.
This is what I want to calculate the measure of: $$lbrace x in X | f_a(x) notin Y text{ for any } a in A rbrace$$
In other words, I want to calculate the measure of the part of $X$ which never hits $Y$ as the space $X$ is dragged along the curve $A$. It is fine to assume that $mu(X), mu(Y) < infty$ and thus calculate its complement, instead. How the heck do I set this up? Is there an averaging method (a la Bogolyubov-Krillov) we can use? It is also fine to assume that $X$ and $Y$ are embedded $n$-manifolds with boundary, so in particular you could alternatively assume that $X, Y$ are open, or regular closed, etc. In other words, they're nice. Locally connected, simply connected, pretty much anything is fine.
Thanks a lot!
real-analysis euclidean-geometry lebesgue-integral geometric-measure-theory
real-analysis euclidean-geometry lebesgue-integral geometric-measure-theory
edited Nov 21 '18 at 10:14
John B
12.2k51840
asked Nov 20 '18 at 23:02
John Samples
1,086516
edited Nov 21 '18 at 10:14
John B
12.2k51840
asked Nov 20 '18 at 23:02
John Samples
1,086516
edited Nov 21 '18 at 10:14
John B
12.2k51840
edited Nov 21 '18 at 10:14
John B
12.2k51840
edited Nov 21 '18 at 10:14
John B
12.2k51840
12.2k51840
asked Nov 20 '18 at 23:02
John Samples
1,086516
asked Nov 20 '18 at 23:02
John Samples
1,086516
asked Nov 20 '18 at 23:02
John Samples
1,086516
1,086516
Do we know the measure of $X$ and $Y$?
– Santana Afton
Nov 20 '18 at 23:43
Just some finite number, all measures are real. In the set-up for the application I have in mind, you would know the measure of $X$ and $Y$ at the outset; no inverse problem. I am looking for a method (probably some integral equation) that works for two fixed $X, Y$, doesn't have to be germane to every pair of subspaces simultaneously.
– John Samples
Nov 21 '18 at 1:04
add a comment |
Do we know the measure of $X$ and $Y$?
– Santana Afton
Nov 20 '18 at 23:43
Just some finite number, all measures are real. In the set-up for the application I have in mind, you would know the measure of $X$ and $Y$ at the outset; no inverse problem. I am looking for a method (probably some integral equation) that works for two fixed $X, Y$, doesn't have to be germane to every pair of subspaces simultaneously.
– John Samples
Nov 21 '18 at 1:04
Do we know the measure of $X$ and $Y$?
– Santana Afton
Nov 20 '18 at 23:43
Do we know the measure of $X$ and $Y$?
– Santana Afton
Nov 20 '18 at 23:43
Just some finite number, all measures are real. In the set-up for the application I have in mind, you would know the measure of $X$ and $Y$ at the outset; no inverse problem. I am looking for a method (probably some integral equation) that works for two fixed $X, Y$, doesn't have to be germane to every pair of subspaces simultaneously.
– John Samples
Nov 21 '18 at 1:04
Just some finite number, all measures are real. In the set-up for the application I have in mind, you would know the measure of $X$ and $Y$ at the outset; no inverse problem. I am looking for a method (probably some integral equation) that works for two fixed $X, Y$, doesn't have to be germane to every pair of subspaces simultaneously.
– John Samples
Nov 21 '18 at 1:04
add a comment |
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Do we know the measure of $X$ and $Y$?
– Santana Afton
Nov 20 '18 at 23:43
Just some finite number, all measures are real. In the set-up for the application I have in mind, you would know the measure of $X$ and $Y$ at the outset; no inverse problem. I am looking for a method (probably some integral equation) that works for two fixed $X, Y$, doesn't have to be germane to every pair of subspaces simultaneously.
– John Samples
Nov 21 '18 at 1:04