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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.