Let . Give examples of sequences and in such that
i) almost everywhere but .
ii) but does not exist for any .
For i), try .
ii) is trickier. You need to use something like the Haar functions.
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?
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).