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If $f$ is a Lebesgue measurable function on $R^n$, we define $K(t)=\lambda\{x\in R^n:|f(x)|>t\}.$ I want to prove that

  1. $\int_0^{\infty}K(t)dt=\int_{R^n} |f(x)|dx$

  2. If $ f\in L^1 (R^n)$, then $\lim_{s\rightarrow t^-} K(s)=K(t).$

I have no idea about these two problems.

1 Answers 1

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I will give hints:

  1. Use Fubini theorem for non-negative functions and $F(x,t):=\chi_S(x,t)$, where $S:=\{(x,t)\in\Bbb R^2,|f(x)|>t\}$.

  2. If $f$ is integrable, then $\{K(s)\}_s$ is decreasing and consists of sets of finite measure. This works, as noted in the comments, if $K(t):=\lambda\{x,|f(x)|\color{red}{\geqslant} t\}$. Otherwise, we take $n=1$, $f=\chi_{(0,1)}$ and $t=1$.

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    It seems (or there is just a typo).2012-12-07