Measure of Subspace Swept Out by Another
up vote
2
down vote
favorite
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 20 hours ago
John B
12.1k51740
asked yesterday
John Samples
1,081516
add a comment |
up vote
2
down vote
favorite
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 20 hours ago
John B
12.1k51740
asked yesterday
John Samples
1,081516
Do we know the measure of $X$ and $Y$?
– Santana Afton
yesterday
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
yesterday
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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 20 hours ago
John B
12.1k51740
asked yesterday
John Samples
1,081516
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 20 hours ago
John B
12.1k51740
asked yesterday
John Samples
1,081516
edited 20 hours ago
John B
12.1k51740
asked yesterday
John Samples
1,081516
edited 20 hours ago
John B
12.1k51740
edited 20 hours ago
John B
12.1k51740
edited 20 hours ago
John B
12.1k51740
12.1k51740
asked yesterday
John Samples
1,081516
asked yesterday
John Samples
1,081516
asked yesterday
John Samples
1,081516
1,081516
Do we know the measure of $X$ and $Y$?
– Santana Afton
yesterday
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
yesterday
add a comment |
Do we know the measure of $X$ and $Y$?
– Santana Afton
yesterday
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
yesterday
Do we know the measure of $X$ and $Y$?
– Santana Afton
yesterday
Do we know the measure of $X$ and $Y$?
– Santana Afton
yesterday
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
yesterday
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
yesterday
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007026%2fmeasure-of-subspace-swept-out-by-another%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Do we know the measure of $X$ and $Y$?
– Santana Afton
yesterday
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
yesterday