I am working on exam, I've failed at 3.2
There is one task, I'd like to ask help about.
Question is: Suppose $ f:\mathbb{R}^n \rightarrow \mathbb{R}$ is integrable.
Prove, that $$\lim_{t\to\infty} tm(\{x\in\mathbb{R}^n:|f(x)|>t\})=0$$
Is it possible to prove, if function is just measurable, not integrable?
My take.. well.. i can scan paper probably.
Dont really understand what i was implied in my answer.
here: https://drive.google.com/open?id=1Dg92XGZVd6w9QG5E4ZNItju2kK8vJ8nG
advises on other tasks are welcome to.
1 i was able to grind.
No it's not… take $f(x) = |x|$ which is measurable.
To proof the statement: You have: $$\int_{\Bbb R} |f| \,dm = \int_{\{|f| \le t\}} |f| \,dm + \int_{\{|f| > t\}} |f| \,dm \ge \int_{\{|f| \le t\}} |f| \,dm + tm(\{x\in\mathbb{R}^n:|f(x)|>t\})$$ hence $$tm(\{x\in\mathbb{R}^n:|f(x)|>t\}) \le \int_{\Bbb R} |f| \,dm - \int_{\{|f| \le t\}} |f| \,dm$$ Now take $\lim$ on both sides…