Absolute value & integrability

  • Thread starter steven187
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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
 

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.
 
176
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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:

quasar987

Science Advisor
Homework Helper
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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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