Let . Give examples of sequences and in such that

i) almost everywhere but .

ii) but does not exist for any .

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- March 7th 2009, 02:54 AMHenryBL^p spaces
Let . Give examples of sequences and in such that

i) almost everywhere but .

ii) but does not exist for any . - March 7th 2009, 03:45 AMOpalg
For i), try .

ii) is trickier. You need to use something like the Haar functions. - March 7th 2009, 02:02 PMHenryB
Thanks a lot. (i) is a good answer but is there really nothing simpler for (ii)? It doesn't seem like the sort of solution that you can really spot yourself.

The question goes on to say:

"For each fnd a subset of [0,1] such that uniformly on \ .

Now it's clearly not what the question is after (I don't actually know what the question is trying to say) but can't we just take for all ?

Secondly, can we find a subsequence such that almost everywhere? - March 8th 2009, 09:53 AMOpalg
The difficulty with (ii) is that to ensure you want g_n to be small over most of the interval; but to prevent existing

*at every point of the interval*you need to arrange that, for every x, g_n(x) stays well away from 0 for infinitely many values of n. The only convenient way I know to achieve that is to define g_n to be 0 on most of the interval, but to bob up to 1 on a small subinterval that gradually works its way along the whole interval.

The easiest way to do this is to construct g_n for n in batches of length . For , where , define to be 1 on the interval , and 0 on the remainder of the interval [0,1].

For the example that I gave, the only point at which is x=1. The sequence does not converge uniformly on [0,1], or even on [0,1), but it does converge to 0 uniformly on , for all .

For the above sequence (g_n), the subsequence converges to 0 almost everywhere (in fact, everywhere except at x=0). - March 9th 2009, 04:49 AMHenryB
Great help - thanks a lot indeed.