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 $varepsilon m(A)$.







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    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 $varepsilon m(A)$.







    share|cite|improve this question






















      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 $varepsilon m(A)$.







      share|cite|improve this question












      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 $varepsilon m(A)$.









      share|cite|improve this question











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      asked Aug 25 at 9:39









      Peter

      547113




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          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$






          share|cite|improve this answer




















          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19










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          1 Answer
          1






          active

          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          1
          down vote













          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$






          share|cite|improve this answer




















          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19














          up vote
          1
          down vote













          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$






          share|cite|improve this answer




















          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19












          up vote
          1
          down vote










          up vote
          1
          down vote









          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$






          share|cite|improve this answer












          Hint



          $$aleq biff forall varepsilon>0, a<b+varepsilon iff exists Asubset mathbb R^N : forall varepsilon>0, a <b+varepsilon m(A).$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 25 at 9:41









          Surb

          36.6k84376




          36.6k84376











          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19
















          • Thank you but I really don't see in what it helps...
            – Peter
            Aug 25 at 10:02










          • What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
            – Surb
            Aug 25 at 12:19















          Thank you but I really don't see in what it helps...
          – Peter
          Aug 25 at 10:02




          Thank you but I really don't see in what it helps...
          – Peter
          Aug 25 at 10:02












          What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
          – Surb
          Aug 25 at 12:19




          What about if $a=m(varphi(A)$ and $b=int_A|det varphi'|$ ?
          – Surb
          Aug 25 at 12:19

















           

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