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Natural way of thinking of the definitions of the rectifiable sets and purely unrectifiable sets
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I'm selfstudy Geometric Measure Theory by Frank Morgan's book and he define rectifiable sets as follows
A set $E subset mathbb{R}^n$ is called $(mathscr{H}^m,m)$ rectifiable if $mathscr{H}^m(E) < infty$ and $mathscr{H}^m$ almost all of $E$ is contained in the union of the images of countably many Lipschitz functions from $mathbb{R}^m$ to $mathbb{R}^n$,
while a $m$-purely unrectifiable set is defined as a set $B$ which has $mathscr{I}^m(B) = 0$, where $mathscr{I}^m$ denote the $m$-dimensional integral-geometric measure.
I found other definition given in Matilla's book, which seems to me slightly different from the Morgan's definition:
A set $E subset mathbb{R}^n$ is called $m$-rectifiable if there is Lipschitz maps $f_i: mathbb{R}^m longrightarrow mathbb{R}^n$ such that
$$mathscr{H}^m left( E backslash bigcup_{i=1}^{infty} f_i(mathbb{R}^m) right) =0,$$
while Matilla define purely unrectifiable set as follows
A set $F subset mathbb{R}^n$ is called purely $m$-unrectifiable if $mathscr{H}^m(E cap F) = 0$ for every $m$-rectifiable set $E$.
My doubts are
$textbf{1.}$ What is the intuition behind the definition of purely $m$-unrectifiable sets?
Morgan and Matilla's definitions are not geometric for me and I can't even realize that are equivalents.
$textbf{2.}$ Is there a natural way of thinking of rectifiable and purely unrectifiable sets?
I can see that both definitions for rectifiable sets generalize the definition of rectifiable curves and I know that the motivation for definition of rectifiable sets is that we want generalize the definition of smooth surfaces in order to allow a kind of rectifiable surface and that we want that $E$ must be contained in the union of countably many Lipschitz function at $mathscr{H}^m$ almost every point of $E$ to ensure that total area remains finite, but I can't see a natural way of thinking of the definition of rectifiable set. I would like to know if there is a natural way of thinking of these definitions, i.e., a way to define a rectifiable set and a purely unrectifiable set having as a start point the motivation for these definitions, for example, this lecture notes has a great explanation how to think a natural way to define Hausdorff measure, explaining the motivation for the definition and pointing out the problems and care we should take with the definition by examples, so I'm looking for an explanation for the definitions of rectifiable sets and purely unrectifiable sets as intuitive as possible.
$textbf{P.S.:}$ Matilla gives a preliminary section on page $202$ and $203$ to motivate the definition of rectifiable sets considering the properties of tangents of the set $E$, but I don't understand.
Thanks in advance!
measure-theory definition intuition geometric-measure-theory
asked Jan 25 at 12:29
GeorgeGeorge
850615
$endgroup$
add a comment |
$begingroup$
I'm selfstudy Geometric Measure Theory by Frank Morgan's book and he define rectifiable sets as follows
A set $E subset mathbb{R}^n$ is called $(mathscr{H}^m,m)$ rectifiable if $mathscr{H}^m(E) < infty$ and $mathscr{H}^m$ almost all of $E$ is contained in the union of the images of countably many Lipschitz functions from $mathbb{R}^m$ to $mathbb{R}^n$,
while a $m$-purely unrectifiable set is defined as a set $B$ which has $mathscr{I}^m(B) = 0$, where $mathscr{I}^m$ denote the $m$-dimensional integral-geometric measure.
I found other definition given in Matilla's book, which seems to me slightly different from the Morgan's definition:
A set $E subset mathbb{R}^n$ is called $m$-rectifiable if there is Lipschitz maps $f_i: mathbb{R}^m longrightarrow mathbb{R}^n$ such that
$$mathscr{H}^m left( E backslash bigcup_{i=1}^{infty} f_i(mathbb{R}^m) right) =0,$$
while Matilla define purely unrectifiable set as follows
A set $F subset mathbb{R}^n$ is called purely $m$-unrectifiable if $mathscr{H}^m(E cap F) = 0$ for every $m$-rectifiable set $E$.
My doubts are
$textbf{1.}$ What is the intuition behind the definition of purely $m$-unrectifiable sets?
Morgan and Matilla's definitions are not geometric for me and I can't even realize that are equivalents.
$textbf{2.}$ Is there a natural way of thinking of rectifiable and purely unrectifiable sets?
I can see that both definitions for rectifiable sets generalize the definition of rectifiable curves and I know that the motivation for definition of rectifiable sets is that we want generalize the definition of smooth surfaces in order to allow a kind of rectifiable surface and that we want that $E$ must be contained in the union of countably many Lipschitz function at $mathscr{H}^m$ almost every point of $E$ to ensure that total area remains finite, but I can't see a natural way of thinking of the definition of rectifiable set. I would like to know if there is a natural way of thinking of these definitions, i.e., a way to define a rectifiable set and a purely unrectifiable set having as a start point the motivation for these definitions, for example, this lecture notes has a great explanation how to think a natural way to define Hausdorff measure, explaining the motivation for the definition and pointing out the problems and care we should take with the definition by examples, so I'm looking for an explanation for the definitions of rectifiable sets and purely unrectifiable sets as intuitive as possible.
$textbf{P.S.:}$ Matilla gives a preliminary section on page $202$ and $203$ to motivate the definition of rectifiable sets considering the properties of tangents of the set $E$, but I don't understand.
Thanks in advance!
measure-theory definition intuition geometric-measure-theory
asked Jan 25 at 12:29
GeorgeGeorge
850615
$endgroup$
add a comment |
$begingroup$
I'm selfstudy Geometric Measure Theory by Frank Morgan's book and he define rectifiable sets as follows
A set $E subset mathbb{R}^n$ is called $(mathscr{H}^m,m)$ rectifiable if $mathscr{H}^m(E) < infty$ and $mathscr{H}^m$ almost all of $E$ is contained in the union of the images of countably many Lipschitz functions from $mathbb{R}^m$ to $mathbb{R}^n$,
while a $m$-purely unrectifiable set is defined as a set $B$ which has $mathscr{I}^m(B) = 0$, where $mathscr{I}^m$ denote the $m$-dimensional integral-geometric measure.
I found other definition given in Matilla's book, which seems to me slightly different from the Morgan's definition:
A set $E subset mathbb{R}^n$ is called $m$-rectifiable if there is Lipschitz maps $f_i: mathbb{R}^m longrightarrow mathbb{R}^n$ such that
$$mathscr{H}^m left( E backslash bigcup_{i=1}^{infty} f_i(mathbb{R}^m) right) =0,$$
while Matilla define purely unrectifiable set as follows
A set $F subset mathbb{R}^n$ is called purely $m$-unrectifiable if $mathscr{H}^m(E cap F) = 0$ for every $m$-rectifiable set $E$.
My doubts are
$textbf{1.}$ What is the intuition behind the definition of purely $m$-unrectifiable sets?
Morgan and Matilla's definitions are not geometric for me and I can't even realize that are equivalents.
$textbf{2.}$ Is there a natural way of thinking of rectifiable and purely unrectifiable sets?
I can see that both definitions for rectifiable sets generalize the definition of rectifiable curves and I know that the motivation for definition of rectifiable sets is that we want generalize the definition of smooth surfaces in order to allow a kind of rectifiable surface and that we want that $E$ must be contained in the union of countably many Lipschitz function at $mathscr{H}^m$ almost every point of $E$ to ensure that total area remains finite, but I can't see a natural way of thinking of the definition of rectifiable set. I would like to know if there is a natural way of thinking of these definitions, i.e., a way to define a rectifiable set and a purely unrectifiable set having as a start point the motivation for these definitions, for example, this lecture notes has a great explanation how to think a natural way to define Hausdorff measure, explaining the motivation for the definition and pointing out the problems and care we should take with the definition by examples, so I'm looking for an explanation for the definitions of rectifiable sets and purely unrectifiable sets as intuitive as possible.
$textbf{P.S.:}$ Matilla gives a preliminary section on page $202$ and $203$ to motivate the definition of rectifiable sets considering the properties of tangents of the set $E$, but I don't understand.
Thanks in advance!
measure-theory definition intuition geometric-measure-theory
asked Jan 25 at 12:29
GeorgeGeorge
850615
$endgroup$
I'm selfstudy Geometric Measure Theory by Frank Morgan's book and he define rectifiable sets as follows
A set $E subset mathbb{R}^n$ is called $(mathscr{H}^m,m)$ rectifiable if $mathscr{H}^m(E) < infty$ and $mathscr{H}^m$ almost all of $E$ is contained in the union of the images of countably many Lipschitz functions from $mathbb{R}^m$ to $mathbb{R}^n$,
while a $m$-purely unrectifiable set is defined as a set $B$ which has $mathscr{I}^m(B) = 0$, where $mathscr{I}^m$ denote the $m$-dimensional integral-geometric measure.
I found other definition given in Matilla's book, which seems to me slightly different from the Morgan's definition:
A set $E subset mathbb{R}^n$ is called $m$-rectifiable if there is Lipschitz maps $f_i: mathbb{R}^m longrightarrow mathbb{R}^n$ such that
$$mathscr{H}^m left( E backslash bigcup_{i=1}^{infty} f_i(mathbb{R}^m) right) =0,$$
while Matilla define purely unrectifiable set as follows
A set $F subset mathbb{R}^n$ is called purely $m$-unrectifiable if $mathscr{H}^m(E cap F) = 0$ for every $m$-rectifiable set $E$.
My doubts are
$textbf{1.}$ What is the intuition behind the definition of purely $m$-unrectifiable sets?
Morgan and Matilla's definitions are not geometric for me and I can't even realize that are equivalents.
$textbf{2.}$ Is there a natural way of thinking of rectifiable and purely unrectifiable sets?
I can see that both definitions for rectifiable sets generalize the definition of rectifiable curves and I know that the motivation for definition of rectifiable sets is that we want generalize the definition of smooth surfaces in order to allow a kind of rectifiable surface and that we want that $E$ must be contained in the union of countably many Lipschitz function at $mathscr{H}^m$ almost every point of $E$ to ensure that total area remains finite, but I can't see a natural way of thinking of the definition of rectifiable set. I would like to know if there is a natural way of thinking of these definitions, i.e., a way to define a rectifiable set and a purely unrectifiable set having as a start point the motivation for these definitions, for example, this lecture notes has a great explanation how to think a natural way to define Hausdorff measure, explaining the motivation for the definition and pointing out the problems and care we should take with the definition by examples, so I'm looking for an explanation for the definitions of rectifiable sets and purely unrectifiable sets as intuitive as possible.
$textbf{P.S.:}$ Matilla gives a preliminary section on page $202$ and $203$ to motivate the definition of rectifiable sets considering the properties of tangents of the set $E$, but I don't understand.
Thanks in advance!
measure-theory definition intuition geometric-measure-theory
measure-theory definition intuition geometric-measure-theory
asked Jan 25 at 12:29
GeorgeGeorge
850615
asked Jan 25 at 12:29
GeorgeGeorge
850615
asked Jan 25 at 12:29
GeorgeGeorge
850615
asked Jan 25 at 12:29
GeorgeGeorge
850615
asked Jan 25 at 12:29
GeorgeGeorge
850615
850615
add a comment |
add a comment |
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