Results 1 to 3 of 3
Like Tree1Thanks
  • 1 Post By Plato

Math Help - Proving Integrability

  1. #1
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Proving Integrability

    The lecturer proved that The drichlet function is not integrable. However I don't find his proof to be correct(Or I don't fully understand the subject, yet). Because he just said that let there be a Partition P and he defined the points, showed that the upper and lower integrations are different from each other. However he uses the same partition for both. But shouldn't there be other conditions? Such as when the partition consist of the rational points, when it consists of irrational points and when it consists of rational and irrational points. He used the same partition for U and L.

    And secondly , the condition of integrability is
    L={L(f,P}|P is partition for the closed interval [a,b]}
    U={U{f,P}|P is partition for the closed interval [a,b]}

    supL<infU (less or equal)

    A function is integrable when supL=infU

    So how come chosing randomly a partition prove the above?

    Thank you.

    Note: I uploaded the proof
    Attached Thumbnails Attached Thumbnails Proving Integrability-drichle-integrability.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,680
    Thanks
    1618
    Awards
    1

    Re: Proving Integrability

    Quote Originally Posted by davidciprut View Post
    he upper and lower integrations are different from each other. However he uses the same partition for both. But shouldn't there be other conditions? Such as when the partition consist of the rational points, when it consists of irrational points and when it consists of rational and irrational points. He used the same partition for U and L.
    And secondly , the condition of integrability is
    L={L(f,P}|P is partition for the closed interval [a,b]}
    U={U{f,P}|P is partition for the closed interval [a,b]}
    supL<infU (less or equal)
    A function is integrable when supL=infU
    So how come chosing randomly a partition prove the above?
    Note: I uploaded the proof
    The uploaded proof is correct.
    Given any partition whatsoever, the $\mathfrak{U}([a,b],P)=b-a~\&~\mathfrak{L}([a,b],P)=0$
    Thanks from davidciprut
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Re: Proving Integrability

    Yes now I understand why. Now I understood the definition of mk(xk-xk-1) ... The mk stands for the inf of the values of f(x) in interval xk-xk-1 and there is definitely a irrational there from the density of irrationals and therefore it has to be equal to zero... Therefore L([a,b],P)=0 for every P.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 17th 2010, 08:30 AM
  2. integrability
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 3rd 2010, 01:57 AM
  3. Integrability
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: May 9th 2010, 06:22 AM
  4. Replies: 1
    Last Post: January 16th 2010, 02:31 AM
  5. proving the failure of riemann integrability
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 10th 2008, 07:43 PM

Search Tags


/mathhelpforum @mathhelpforum