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Topic: integral over difference of sets
Replies: 4   Last Post: Jul 3, 2013 12:21 PM

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umumenu@gmail.com

Posts: 910
Registered: 3/9/08
Re: integral over difference of sets
Posted: Nov 9, 2010 2:51 AM
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On Nov 8, 8:17 pm, Arturo Magidin <magi...@member.ams.org> wrote:
> On Nov 8, 12:37 pm, Anonymous wrote:
>

> > Hello everyone,
> > I know that the integral of a simple function over the sum of two sets
> > is the integral of the function over each individual set.  (Additivity
> > for simple functions).

>
> If both sets are measurable...


Redundant.

> >  My question is if B is a subset of A.  Is the
> > integral over the difference (A-B) the difference of the integrals,
> > i.e the difference of the integtral of the function over A minus the
> > integral of the function over B.

>
> > To put it another way does sum(ai*  x m{x in A-B|
> > yi-1<=f(x)<yi})=sum(ai*  x m{x in A| yi-1<=f(x)<yi}-sum(ai*  x m{x in
> > B| yi-1<=f(x)<yi})?

>
> Since A = (A-B)\/B, if both A and B are measurable, you can use the
> first statement you say you know, apply it to (A-B)\/B, and then
> "solve" for the integral over A-B.


Han de Bruijn



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