Why if $m(varphi(A))leq int_A|det varphi'|$ doesn't hold, there is $varepsilon>0$ s.t. $m(varphi(A))>int_A|det varphi'|+varepsilon m(A)$?
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Let $A$ a closed cube, $m$ the Lebesgue measure on $mathbb R^n$ and $varphi:Uto mathbb R^n$ a $mathcal C^1$ application (but this point is not important for my question). I recal that $varphi'$ is the Jacobian matrix. Why if $$m(varphi(A))leq int_A|det varphi'|$$ doesn't hold for a closed cube of side $a$, then there is $varepsilon>0$ s.t. $$m(varphi(A))>int_A|det varphi'|+varepsilon m(A) ?$$
For me it would be only : there is a cube of side $a$ s.t. $$m(varphi(A))>int_A |det varphi'|,$$
I don't understand the thing with the term $varepsilon m(A)$.
real-analysis
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up vote
-1
down vote
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Let $A$ a closed cube, $m$ the Lebesgue measure on $mathbb R^n$ and $varphi:Uto mathbb R^n$ a $mathcal C^1$ application (but this point is not important for my question). I recal that $varphi'$ is the Jacobian matrix. Why if $$m(varphi(A))leq int_A|det varphi'|$$ doesn't hold for a closed cube of side $a$, then there is $varepsilon>0$ s.t. $$m(varphi(A))>int_A|det varphi'|+varepsilon m(A) ?$$
For me it would be only : there is a cube of side $a$ s.t. $$m(varphi(A))>int_A |det varphi'|,$$
I don't understand the thing with the term $varepsilon m(A)$.
real-analysis
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let $A$ a closed cube, $m$ the Lebesgue measure on $mathbb R^n$ and $varphi:Uto mathbb R^n$ a $mathcal C^1$ application (but this point is not important for my question). I recal that $varphi'$ is the Jacobian matrix. Why if $$m(varphi(A))leq int_A|det varphi'|$$ doesn't hold for a closed cube of side $a$, then there is $varepsilon>0$ s.t. $$m(varphi(A))>int_A|det varphi'|+varepsilon m(A) ?$$
For me it would be only : there is a cube of side $a$ s.t. $$m(varphi(A))>int_A |det varphi'|,$$
I don't understand the thing with the term $varepsilon m(A)$.
real-analysis
Let $A$ a closed cube, $m$ the Lebesgue measure on $mathbb R^n$ and $varphi:Uto mathbb R^n$ a $mathcal C^1$ application (but this point is not important for my question). I recal that $varphi'$ is the Jacobian matrix. Why if $$m(varphi(A))leq int_A|det varphi'|$$ doesn't hold for a closed cube of side $a$, then there is $varepsilon>0$ s.t. $$m(varphi(A))>int_A|det varphi'|+varepsilon m(A) ?$$
For me it would be only : there is a cube of side $a$ s.t. $$m(varphi(A))>int_A |det varphi'|,$$
I don't understand the thing with the term $varepsilon m(A)$.
real-analysis
asked Aug 25 at 9:39
Peter
547113
547113
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1 Answer
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Hint
$$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$
Thank you but I really don't see in what it helps...
â Peter
Aug 25 at 10:02
What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
â Surb
Aug 25 at 12:19
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint
$$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$
Thank you but I really don't see in what it helps...
â Peter
Aug 25 at 10:02
What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
â Surb
Aug 25 at 12:19
add a comment |Â
up vote
1
down vote
Hint
$$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$
Thank you but I really don't see in what it helps...
â Peter
Aug 25 at 10:02
What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
â Surb
Aug 25 at 12:19
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Hint
$$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$
Hint
$$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$
answered Aug 25 at 9:41
Surb
36.6k84376
36.6k84376
Thank you but I really don't see in what it helps...
â Peter
Aug 25 at 10:02
What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
â Surb
Aug 25 at 12:19
add a comment |Â
Thank you but I really don't see in what it helps...
â Peter
Aug 25 at 10:02
What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
â Surb
Aug 25 at 12:19
Thank you but I really don't see in what it helps...
â Peter
Aug 25 at 10:02
Thank you but I really don't see in what it helps...
â Peter
Aug 25 at 10:02
What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
â Surb
Aug 25 at 12:19
What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
â Surb
Aug 25 at 12:19
add a comment |Â
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