Re: Question about Lebesque Integral

On 2005-05-03, agapito6314@xxxxxxx <agapito6314@xxxxxxx> wrote:
> The Lebesque integral ,I, of function f with respect to measure u is
> defined by
>
> I (f du) = I (f+ du) - I (f- du)
>
> where f+ and f- are the positive and negative parts of f, respectively.
> Now my text (Bartle) states that "it is easy to see that if" f = v -
> w, with v and w non-negative, then
>
> I (f du)= I (v du) - I ( w du)
>
> I don't see how this follows from the basic definition of I as supremum
> of integrals of simple functions. Can someone please help with a proof?

You can't. The definition using suprema of integrals of simple functions
works only if f itself is non-negative. To show the second equation,
split f into its positive and negative parts (where is v greater than w,
and vice versa?), them apply the definition for signed integrands (the
first equation).

Hope this helps.

Igor
.

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