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Absolute value & integrability

  1. Jun 22, 2005 #1
    hello all

    Iv been working on alot of integrability questions and im having trouble with this problem
    let f be integrable on [a,b] then show that |f| is integrable and that

    [tex]|\int_{a}^{b}f|\le \int_{a}^{b}|f|[/tex]

    now this is what i know

    [tex]\int_{a}^{b^U}f =\int_{a_{L}}^{b}f= \int_{a}^{b}f[/tex]

    [tex] U(f,P)-L(f,P)<\epsilon[/tex]

    and

    [tex]|f(x)|\le M \forall x\epsilon [a,b][/tex] is there anything else i can gain from a function being integrable on a closed interval?

    muchly appreciated if someone could tell me where to start and some directions? I realise that it is only through practice that i will be able to know where to start and where to go from there, please help

    thank you

    steven
     
  2. jcsd
  3. Jun 22, 2005 #2

    quasar987

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    Hi,

    You need to show that for a given partition P, S(|f|,P) - s(|f|,P) [itex]\leq[/itex] S(f,P) - s(f,P).

    It is easy: use the definition of s(,) and S(,) and work the three different cases for a given interval in the partition: 1) f(x) is stricly < 0 for all x in that interval. 2) f(x) is stricly > 0 for all x in that interval. 3) f(x) is < 0 for some x and > 0 for some other x in that interval.
     
  4. Jun 22, 2005 #3
    hello all

    this is what i have done so far, i hope it is correct, i have shown that
    [tex] U(|f|,P)-L(|f|,P)<\epsilon[/tex]
    and so |f| is integrable that wasnt a problem
    then since -|f(x)|<=f(x)<=|f(x)| for all x an element of [a,b]
    then we integrate the whole inequality to get
    [tex] -\int_{a}^{b}|f(x)| \le\int_{a}^{b}f(x)\le\int_{a}^{b}|f(x)| [/tex]
    and hence
    [tex]|\int_{a}^{b}f|\le \int_{a}^{b}|f|[/tex]

    In terms of the above method about proving the 3 different cases i got pretty confused going down that path, some further details would be helpful

    steven
     
    Last edited: Jun 22, 2005
  5. Jun 22, 2005 #4

    quasar987

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    How about simply invoquing the caracterisation of the integral

    [tex]\int_{a}^{b}f(x)dx = \lim_{|p|\rightarrow 0}\sum_{i=1}^{n}f(t_i)(x_i-x_{i-1})[/tex]

    and the triangle inequality:

    [tex]\forall x,y \in \mathbb{R}, \ |x+y| \leq |x|+|y|[/tex]

    ?
     
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