# Littlewood-like principles

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