Originally Posted by

**mgarson** Hi,

I have a bdd Lebesgue measurable function $\displaystyle h$ on $\displaystyle \mathbb{R}$ such that h(x)=h(x+1) a.e. on $\displaystyle \mathbb{R}$. Define $\displaystyle h_k(x)=h(kx)$.

How do I prove the following:

(i) $\displaystyle \lim_{k\rightarrow\infty}\int_a^bh_k(x)dx=(b-a)\int_0^1h(x)dx$

(ii) If $\displaystyle h$ isn't constant a.e. then there will be no subseq that will converge $\displaystyle m$-a.e. on $\displaystyle \[a,b\]$.

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My attempts:

(i) For this one I'm thinking that the condition on $\displaystyle h$ implies that $\displaystyle h$ is constant a.e., say $\displaystyle h=C$ a.e.