Let ,1 and let . Show that there exists a set with such that if and ,then

I let ,where and .

I am not sure whether my construction of the sets are correct because I can not make any conclusion regarding .

Can anyone comment on this?

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- Jan 29th 2010, 06:37 AMproblemLebesgue Integral
Let ,1 and let . Show that there exists a set with such that if and ,then

I let ,where and .

I am not sure whether my construction of the sets are correct because I can not make any conclusion regarding .

Can anyone comment on this? - Jan 29th 2010, 01:05 PMOpalg
I think you have to go right back to the definition of the Lebesgue integral to do this properly. For a positive function such as , the usual way to define its integral (as described here, for example) is that it is the supremum of the integrals of nonnegative simple functions majorised by . If the integral is finite (as it is if ) then each of these simple functions must have finite integral and therefore finite support. We can find a simple function s such that and . Now take to be the support of s. You should find that this does the required job.