Sunday, 4 September 2016

real analysis - Dilations of Integrable Function Converge to Zero Almost Everywhere

Suppose f:[0,)[0,) is integrable. Set fn(x):=f(nx). I want to show that fn(x)0 almost everywhere or equivalently, the set
{x:lim sup
has measure zero, for any \delta>0.



By dilation invariance, it's clear that f_{n}\rightarrow 0 in L^{1} and therefore also in measure. Furthermore, we can pass to a subsequence to obtain a.e. convergence. If f has compact support, then it's obvious that f_{n}\rightarrow 0 almost everywhere.



My thought was to try approximating f in L^{1} by g\in C_{c}(\mathbb{R}) and use something like



|\{\limsup f_{n}\geq\delta\}|\leq|\{\limsup|f_{n}-g_{n}|\geq\delta/2\}|+|\{\limsup|g_{n}|\geq\delta/2\}|




and go from there. But I'm not sure how to control the first term on the RHS. Any suggestions?

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