Suppose i have f is an odd function.

How to show that =0 ?

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- Dec 11th 2011, 10:24 AMyounhockLebesgue integral of Odd Function.
Suppose i have f is an odd function.

How to show that =0 ? - Dec 11th 2011, 10:35 AMchisigmaRe: Lebesgue integral of Odd Function.
- Dec 11th 2011, 07:04 PMyounhockRe: Lebesgue integral of Odd Function.
- Dec 11th 2011, 10:15 PMchisigmaRe: Lebesgue integral of Odd Function.
An odd function that is not the null function is positive in a part and negative in a different part of the definition interval. Let's define where and elsewhere and where and elsewhere. The Lebesgue integral of f(x) is defined as...

(1)

Now the integral (1) exists only if both the integrals in (1) exist and that is not true [for example...] for in ...

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- Dec 11th 2011, 10:35 PMDrexel28Re: Lebesgue integral of Odd Function.
Well, isn't Lebesgue integrable on , so it's a moot point, no? The basic idea is that change of variables still works for Lebsgue integrals. What if you let ?

- Dec 11th 2011, 10:47 PMchisigmaRe: Lebesgue integral of Odd Function.
I'm very sorry but I'm afraid not to have fully undestood Your reply... can You explain a little more, please?...

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- Dec 11th 2011, 10:51 PMDrexel28Re: Lebesgue integral of Odd Function.
- Dec 11th 2011, 11:06 PMchisigmaRe: Lebesgue integral of Odd Function.
The original question was to demonstrate that if f(x) is and odd function, then ...

(1)

... where the integral is 'Lebesgue Integral'. My replay has been [symply...] that an odd function isn't necessarly Lebesgue integrable and the example has been supplied. Of course if we add the condition that f(x) must be 'regular', then the demonstration of (1) is very comfortable...

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- Dec 11th 2011, 11:15 PMDrexel28Re: Lebesgue integral of Odd Function.
- Dec 12th 2011, 02:30 AMyounhockRe: Lebesgue integral of Odd Function.
Maybe I should add one more condition to the question since it causes confusion. Now f(x) is lebesgue integrable and odd. how to show the integral value is zero?

- Dec 12th 2011, 04:23 AMchisigmaRe: Lebesgue integral of Odd Function.
... if f(x) is odd then so that is...

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