The continuous measure function

Question

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.

Answer 1

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 $t=1$.