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 ...
Marry Christmas from Serbia
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...
Marry Christmas from Serbia