# Lorentz space

1. Sep 1, 2010

### 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" $$f^{*}$$:

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

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!

Last edited: Sep 1, 2010