Distribution function
Let 
, and define the distribution function of 
,
.
It has the following properties:
-a.e. implies that 
for all 
-
-
Here’s a neat result from Grafakos’ ‘Classical and Modern Fourier Analysis’: if 
is increasing and 
, then
Actually, I think you also need to state 
. As a special case,
.
The proof consists of writing 
as an integral in terms of 
, extending the limits of integration of that integral to 
by inserting an appropriate characteristic function 
, 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 
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, is the same for all translates of ), yet it determines the value of these particular functionals at 
.
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}.](/cz/latexrender/pictures/d66ea28bf52b40ce2954a41d79066380.png "\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 
. A non-trivial result!
Possibly relevant posts:
- Fourier HW (6/16/2005)
- Radial filters preserved under convolution? (8/15/2006)
- Chebyshev’s inequality (continuous, general version) (1/15/2006)
[…] Oh, and I did figure out how to prove that heart stopper: use Fubini. I actually have seen this before, in a different context, when I was taking Papadakis’ fourier analysis course. See what I mean about forgetting stuff? […]