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        general >> wanted >> measure theory ....
        
(Message started by: trusure on Oct 12^th, 2009, 7:08pm)

Title: measure theory ....
Post by trusure on Oct 12^th, 2009, 7:08pm

I just proved the following result:
if
(1) f_n is a sequence which converges to f a.e.,
(2) f_n>=0, and lebesgue measurable on R.
(3) integral(f) is finite, and
(4) integral (f_n) converges to integral (f) then:

integral (f_n) converges to integral (f) over any measurable set A.

(note: the integration is Lebesgue integral)

Now, I'm looking for a counterexample that this result is not true if integral (f) is infinity.

so.... any one can help me ....

thanks for helping in advance

Title: Re: measure theory ....
Post by Obob on Oct 12^th, 2009, 9:30pm

Let f_n(x) = max (1/x²,1/n) on the whole real line.

Title: Re: measure theory ....
Post by trusure on Oct 12^th, 2009, 10:29pm

ok, so we have f_n ---> f=1/x^2,
but integral of f over R is finite =0, ??!!

moreover, what about the set A we will use to get a contradiction ?

Title: Re: measure theory ....
Post by Obob on Oct 13^th, 2009, 5:18am

I'm guessing this is homework, so you need to fill in some of the details. And it is definitely not true that the integral of 1/x^2 over R is finite (let alone zero). How on earth could it be zero? It is a function that is strictly positive everywhere.

There's a couple other examples that fail for the same reason:

Let your measure space X consist of two points p & q, both with infinite measure. Put f_n(p) = 1, f_n(q) = 1/n.

Or let your measure space X be a disjoint union of two smaller spaces A and B, where A has infinite measure and B has positive measure. Let g be a function on B with infinite integral, and define f_n(a) = 1/n for a in A, f_n(b) = g(b) for b in B.

The first example I gave can be seen as being essentially equivalent to the second construction here.