Hi,

I have a bdd Lebesgue measurable function on such that h(x)=h(x+1) a.e. on . Define .

How do I prove the following:

(i)

(ii) If isn't constant a.e. then there will be no subseq that will converge -a.e. on .

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My attempts:

(i) For this one I'm thinking that the condition on implies that is constant a.e., say a.e. so

.

But if that is true... the the is of no use.

(ii) Here I'm thinking a contradiction, i.e. assuming that there is such a sub.seq. but I get it to work...