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 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 above sequence (g_n), the subsequence converges to 0 almost everywhere (in fact, everywhere except at x=0).