My question arose while studying an article which finds the $K$-functional for the pair of spaces $L^1,L^\infty$, so it's related to interpolation theory, but I think it can be solved with some $\inf,\sup$ manipulations. I'm sorry if the tags aren't correct.
Consider $(X,\Sigma, \mu)$ an arbitrary measure space. For each measurable function $f: X \to \Bbb{C}$ and $\alpha \geq 0$ define $ f_*(\alpha)=\mu(\{ x \in X : |f(x)|>\alpha \}),$ the distribution function of $f$.
For any measurable function $f:X \to \Bbb{C}$ for which there exists $\alpha>0$ with $f_*(\alpha)<\infty$, define $f^* :(0,\infty) \to [0,\infty)$ by $ f^*(t)=\inf \{ y>0 :f_*(y) \leq t\}.$
It can be proved from the definitions that for every $t>0$ we have $ f^*(f_*(t))\leq t,\ f_*(f^*(t))\leq t.$
Moreover, the function $f^*$ is continuous from the right. Both of the functions $f_*,f^*$ are non increasing.
What I need to prove is that
$\sup_{t>0} f^*(t)= \| f\|_\infty,$ for every function $f \in L^\infty$.
The inequality $\leq $ is straight from the definition. The other one I can't seem to get, and the text says that it's pretty hard, but with "a little more effort" it can be done. I haven't succeeded.