0
$\begingroup$

Let $f: \mathbb R^n \to \mathbb R$ be a measurable fucntion. Define $F(t) = \mu \{x:|f(x)| >t\}$

  1. Show that $F$ is nonincreasing and right-continuous (done).

  2. Define $F^\star(v)=\inf \{t: F(t)\leq v\}$ Need to show that $F^\star: \mathbb R^+\to\mathbb R^+$ is a nonincreasing and right-continuous function.

  3. By using continuity on the right prove $F(F^\star(t))\leq t$ (no idea)

  4. Show that $F(t) = m\{v: F^\star(v)>t\}$ ($m$=Lebesgue measure).

Any help will be appreciated!

1 Answers 1