What is the Definition of the Rearrangement Function f* in Lorentz Space?

[zeebek]

I am reading the definition in wiki ( nothing better at the moment)
http://en.wikipedia.org/wiki/Lorentz_space

It seems too vague for me, namely what they call "rearrangement function" $f^{*}$:

$$f^{*}: [0, \infty) \rightarrow [0, \infty]; \\<br /> <br /> f^{*}(t) = \inf\{\alpha \in \mathbb{R}^{+}: d_f(\alpha) \leq t\}; \\<br /> <br /> d_f(\alpha) = \mu(\{x \in X : |f(x)| > \alpha\}).<br />$$

I am trying to put in words what is written. Is it right:

first for a given $t$ we are looking for all $\alpha$'s, so that $d_f(\alpha) \leq t$, where $d_f(\alpha)$ is basically a size of the area where $|f(x)| > \alpha$? Then we take infinum via $\alpha$, so as a result there will be the smallest $d$?

Still I cannot imagine "geometrically" how is it?

At last, I need just simpler difinition for the case when $f$ is real.

thanks!

 

[fresh_42]

Let's look at an easy example, i.e. one where all quantities are real numbers. We don't start with the infimum, but with the measurable set instead. Given the function $f$ as below, and a level $a$. Then $d_f(a)$ is the pink area. Now we shift $f$ upwards until this area is as big as $t$ and define $f*(t)$ to be this maximal shift.