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Zero content - geometric interpretation
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I don't understand what it means for a set to have zero content.
What is its geometric interpretation?
Can you give me some examples of sets in Rn that do not have any zero contents?
Also, can you clarify the primary relationship between zero content and integration in Rn
calculus integration general-topology multivariable-calculus
edited Jul 18 at 23:42
Andrés E. Caicedo
63.2k7151235
asked Jul 18 at 17:23
Arina Momajjed
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up vote
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I don't understand what it means for a set to have zero content.
What is its geometric interpretation?
Can you give me some examples of sets in Rn that do not have any zero contents?
Also, can you clarify the primary relationship between zero content and integration in Rn
calculus integration general-topology multivariable-calculus
edited Jul 18 at 23:42
Andrés E. Caicedo
63.2k7151235
asked Jul 18 at 17:23
Arina Momajjed
312
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I don't understand what it means for a set to have zero content.
What is its geometric interpretation?
Can you give me some examples of sets in Rn that do not have any zero contents?
Also, can you clarify the primary relationship between zero content and integration in Rn
calculus integration general-topology multivariable-calculus
edited Jul 18 at 23:42
Andrés E. Caicedo
63.2k7151235
asked Jul 18 at 17:23
Arina Momajjed
312
I don't understand what it means for a set to have zero content.
What is its geometric interpretation?
Can you give me some examples of sets in Rn that do not have any zero contents?
Also, can you clarify the primary relationship between zero content and integration in Rn
calculus integration general-topology multivariable-calculus
edited Jul 18 at 23:42
Andrés E. Caicedo
63.2k7151235
asked Jul 18 at 17:23
Arina Momajjed
312
edited Jul 18 at 23:42
Andrés E. Caicedo
63.2k7151235
edited Jul 18 at 23:42
Andrés E. Caicedo
63.2k7151235
edited Jul 18 at 23:42
Andrés E. Caicedo
63.2k7151235
63.2k7151235
asked Jul 18 at 17:23
Arina Momajjed
312
asked Jul 18 at 17:23
Arina Momajjed
312
asked Jul 18 at 17:23
Arina Momajjed
312
312
add a comment |Â
add a comment |Â
1 Answer
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A set with zero content is a subset of $Bbb R^n$ that can be covered by finitely many rectangular $n$-parallelepipeds whose combined size can be chosen as small as you want (the size is the product of the lengths of the sides).
A set with zero content necessarily has Lebesgue measure zero and therefore integration over such sets always outputs zero.
An example of a subset of $Bbb R^n$ without zero content is for instance $[0,1]^n$. If you cover this set by finitely many rectangular parallelepipeds, their combined size has to be at least $1$.
answered Jul 18 at 17:36
Arnaud Mortier
19.1k22159
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
A set with zero content is a subset of $Bbb R^n$ that can be covered by finitely many rectangular $n$-parallelepipeds whose combined size can be chosen as small as you want (the size is the product of the lengths of the sides).
A set with zero content necessarily has Lebesgue measure zero and therefore integration over such sets always outputs zero.
An example of a subset of $Bbb R^n$ without zero content is for instance $[0,1]^n$. If you cover this set by finitely many rectangular parallelepipeds, their combined size has to be at least $1$.
answered Jul 18 at 17:36
Arnaud Mortier
19.1k22159
add a comment |Â
up vote
1
down vote
A set with zero content is a subset of $Bbb R^n$ that can be covered by finitely many rectangular $n$-parallelepipeds whose combined size can be chosen as small as you want (the size is the product of the lengths of the sides).
A set with zero content necessarily has Lebesgue measure zero and therefore integration over such sets always outputs zero.
An example of a subset of $Bbb R^n$ without zero content is for instance $[0,1]^n$. If you cover this set by finitely many rectangular parallelepipeds, their combined size has to be at least $1$.
answered Jul 18 at 17:36
Arnaud Mortier
19.1k22159
add a comment |Â
up vote
1
down vote
up vote
1
down vote
A set with zero content is a subset of $Bbb R^n$ that can be covered by finitely many rectangular $n$-parallelepipeds whose combined size can be chosen as small as you want (the size is the product of the lengths of the sides).
A set with zero content necessarily has Lebesgue measure zero and therefore integration over such sets always outputs zero.
An example of a subset of $Bbb R^n$ without zero content is for instance $[0,1]^n$. If you cover this set by finitely many rectangular parallelepipeds, their combined size has to be at least $1$.
answered Jul 18 at 17:36
Arnaud Mortier
19.1k22159
A set with zero content is a subset of $Bbb R^n$ that can be covered by finitely many rectangular $n$-parallelepipeds whose combined size can be chosen as small as you want (the size is the product of the lengths of the sides).
A set with zero content necessarily has Lebesgue measure zero and therefore integration over such sets always outputs zero.
An example of a subset of $Bbb R^n$ without zero content is for instance $[0,1]^n$. If you cover this set by finitely many rectangular parallelepipeds, their combined size has to be at least $1$.
answered Jul 18 at 17:36
Arnaud Mortier
19.1k22159
answered Jul 18 at 17:36
Arnaud Mortier
19.1k22159
answered Jul 18 at 17:36
Arnaud Mortier
19.1k22159
answered Jul 18 at 17:36
Arnaud Mortier
19.1k22159
19.1k22159
add a comment |Â
add a comment |Â
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