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 Post subject: MeasurabilityPosted: Mon, 23 Mar 2009 10:52:54 UTC
 Senior Member

Joined: Sat, 15 Dec 2007 15:57:40 UTC
Posts: 67
Location: Kyoto
Let be a measurable space with , and let be
an integrable positive function (B is Borel-set), i.e. satisfying

Show that the integral can be calculated as the limit

where for every n\in N

is any number satisfying and

I actually understand the question, I just have no idea how to do it, how to write it down or work it out.

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 Post subject: Re: MeasurabilityPosted: Mon, 23 Mar 2009 14:41:04 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6009
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Sarah wrote:
Let be a measurable space with , and let be
an integrable positive function (B is Borel-set), i.e. satisfying

Show that the integral can be calculated as the limit

where for every n\in N

is any number satisfying and

I actually understand the question, I just have no idea how to do it, how to write it down or work it out.

I think your last condition should be

If f is a simple function the result holds. So approximate f by simple functions ...

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 Post subject: Re: MeasurabilityPosted: Mon, 23 Mar 2009 14:41:44 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sat, 7 Jan 2006 18:29:24 UTC
Posts: 1401
Location: Leeds, UK
Sarah wrote:
Let be a measurable space with , and let be
an integrable positive function (B is Borel-set), i.e. satisfying

Show that the integral can be calculated as the limit

where for every n\in N

is any number satisfying and

I actually understand the question, I just have no idea how to do it, how to write it down or work it out.

Let , where is the function that takes the value 1 on the set and is zero elsewhere. Notice that is integrable, with integral , and that for all x in X. Thus .

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 Post subject: Posted: Mon, 23 Mar 2009 21:36:03 UTC
 Senior Member

Joined: Sat, 15 Dec 2007 15:57:40 UTC
Posts: 67
Location: Kyoto
Of course the last condition is incorrect, thanks for noticing!
The help looks great, I going to write it down completely...

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