Distribution function

January 14th, 2006 ~ Posted in: Mathematics

Let f \in L^p(X, \mu), and define the distribution function of f,

\displaystyle d_f(\alpha) = \mu\left(\left\{x: |f(x)|> \alpha \right\}\right).

It has the following properties:

  1. |g| \leq |f| \mu-a.e. implies that d_g \leq d_f
  2. d_{cf} = \frac{1}{|c|} d_f for all c \in \C \setminus \{0\}
  3. d_{f+g}(\alpha + \beta) \leq d_f(\alpha) + d_g(\beta)
  4. d_{fg}(\alpha \beta) \leq d_f(\alpha) + d_g(\beta)

Here’s a neat result from Grafakos’ ‘Classical and Modern Fourier Analysis’: if \phi : [0, \infty) \rightarrow \R is increasing and C^1, then

\displaystyle \int_X \phi\left(|f|\right) d\mu = \int_0^\infty \phi^\prime(\alpha)d_f(\alpha) d\alpha.

Actually, I think you also need to state \phi(0)=0. As a special case,

\displaystyle\|f\|^p_p = p \int_0^\infty \alpha^{p-1} d_f(\alpha) d\alpha .

The proof consists of writing \phi(|f|) as an integral in terms of \phi^\prime, extending the limits of integration of that integral to 0, \infty by inserting an appropriate characteristic function \chi(x,\alpha), using Fubini’s theorem– then the trickiest part is recognizing that the original characteristic function is exactly the same as the one needed to get d_f as your interior integral. I’m too lazy to write it out, but it’s a good exercise: it took me embarassingly long to figure out how Grafakos applied the Fubini theorem (namely, the equivalence of the characteristic functions).

The interesting thing about the distribution function is that it is not unique to the function (as a trivial example, under Lebesgue measure, d_f is the same for all translates of f), yet it determines the value of these particular functionals at f \in \L^p.

As an example, let

\displaystyle \phi(x) = \begin{cases} \sin(x) & x \in [0, 1] \\ 0 & \text{ otherwise} \end{cases} \quad f(x) = \begin{cases} 1-x^2 & x \in [0,1] \\ 0 & \text { otherwise } \end{cases}.

You can use the above result to show \int_0^1 \sin(1- x^2) dx = \int_0^1 \cos(\alpha) \sqrt{1 - \alpha} d\alpha . A non-trivial result!

Mathematics ~

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