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I am asking on what seems to be a non-trivial variant of the popular questions:

1) if $f\in L^1(\mathbb{R}^d)$ then for all $\epsilon >0$,there exist $R>0$ such that $\int_{\mathbb{R}^d\setminus B(0,R)}|f(x)|dx<\epsilon$

2) If $f:\mathbb{R}^d\rightarrow \mathbb{C}$ is measurable and $m(spt(f))<\infty$, then for all $\epsilon>0$, there exist a measurable set $E\subset \mathbb{R}^d$ of measure at most $\epsilon$ outside of wich f is locally bounded. (i.e for all $R>0$ there exist $M<\infty$ such that $|f(x)|\leq M$ for all $x\in B(0,R)\setminus E $.

I think for 1) use Liitewood's principle for absolutely integrable function and Markov inequality but I'm so confused

any hints? thanks

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