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What is a tube in $mathbb{R}^n$?
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Lebesgue's differentiation theorem states that if $x$ is a point in $mathbb{R}^n$ and $f:mathbb{R}^nrightarrowmathbb{R}$ is a Lebesgue integrable function, then the limit of $frac{int_B f dlambda}{lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles.
But I just came across a journal paper which shows that if you take the collection of all "tubes" in $mathbb{R}^n$ oriented in certain directions, then the Lebesgue differentiation holds true for this collection for $L^p$ functions with $p>1$. But my question is, what exactly is a tube in $mathbb{R}^n$ as the term is used in this paper? The paper doesn't provide any definition as far as I can tell.
Is it like a cylinder, or what?
geometry measure-theory definition lebesgue-integral lebesgue-measure
asked 8 hours ago
Keshav Srinivasan
2,45411340
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Lebesgue's differentiation theorem states that if $x$ is a point in $mathbb{R}^n$ and $f:mathbb{R}^nrightarrowmathbb{R}$ is a Lebesgue integrable function, then the limit of $frac{int_B f dlambda}{lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles.
But I just came across a journal paper which shows that if you take the collection of all "tubes" in $mathbb{R}^n$ oriented in certain directions, then the Lebesgue differentiation holds true for this collection for $L^p$ functions with $p>1$. But my question is, what exactly is a tube in $mathbb{R}^n$ as the term is used in this paper? The paper doesn't provide any definition as far as I can tell.
Is it like a cylinder, or what?
geometry measure-theory definition lebesgue-integral lebesgue-measure
asked 8 hours ago
Keshav Srinivasan
2,45411340
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Lebesgue's differentiation theorem states that if $x$ is a point in $mathbb{R}^n$ and $f:mathbb{R}^nrightarrowmathbb{R}$ is a Lebesgue integrable function, then the limit of $frac{int_B f dlambda}{lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles.
But I just came across a journal paper which shows that if you take the collection of all "tubes" in $mathbb{R}^n$ oriented in certain directions, then the Lebesgue differentiation holds true for this collection for $L^p$ functions with $p>1$. But my question is, what exactly is a tube in $mathbb{R}^n$ as the term is used in this paper? The paper doesn't provide any definition as far as I can tell.
Is it like a cylinder, or what?
geometry measure-theory definition lebesgue-integral lebesgue-measure
asked 8 hours ago
Keshav Srinivasan
2,45411340
Lebesgue's differentiation theorem states that if $x$ is a point in $mathbb{R}^n$ and $f:mathbb{R}^nrightarrowmathbb{R}$ is a Lebesgue integrable function, then the limit of $frac{int_B f dlambda}{lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles.
But I just came across a journal paper which shows that if you take the collection of all "tubes" in $mathbb{R}^n$ oriented in certain directions, then the Lebesgue differentiation holds true for this collection for $L^p$ functions with $p>1$. But my question is, what exactly is a tube in $mathbb{R}^n$ as the term is used in this paper? The paper doesn't provide any definition as far as I can tell.
Is it like a cylinder, or what?
geometry measure-theory definition lebesgue-integral lebesgue-measure
geometry measure-theory definition lebesgue-integral lebesgue-measure
asked 8 hours ago
Keshav Srinivasan
2,45411340
asked 8 hours ago
Keshav Srinivasan
2,45411340
asked 8 hours ago
Keshav Srinivasan
2,45411340
asked 8 hours ago
Keshav Srinivasan
2,45411340
asked 8 hours ago
Keshav Srinivasan
2,45411340
2,45411340
add a comment |
add a comment |
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