Exploring Lorentz Space: Definition of f*

[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" [itex]f^{*}[/itex]:

[tex]f^{*}: [0, \infty) \rightarrow [0, \infty]; \\

f^{*}(t) = \inf\{\alpha \in \mathbb{R}^{+}: d_f(\alpha) \leq t\}; \\

d_f(\alpha) = \mu(\{x \in X : |f(x)| > \alpha\}).

[/tex]

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

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

Still I cannot imagine "geometrically" how is it?

At last, I need just simpler difinition for the case when [itex]f[/itex] 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.