From: Wilbert Dijkhof <w.j.dijkhof@student.utwente.nl>
Newsgroups: sci.math
Subject: Re: problem with lebesgue-borel measures along with fubini's theorem = lots of fun!
Date: Thu, 19 Nov 1998 16:56:54 +0100

Mark Lokhorst wrote:

> Hi,
>
> I need some help on this one from
> whoever *likes* to play with Lebesgue-Borel measures.
>
> Question:
>
> Proof (inf = infinity):
>
>    inf                   inf
>     /                     /
> d   |                     |
> --  | cos(xt) f(t) dt = - | sin(xt) tf(t) dt
> dx  /                     /
>   -inf                  -inf
>
> using Fubini's theorem for each f which has a Lebesgue-Borel measure
> so that
>
>  inf
>   /
>   |
>   | |tf(t)|dt < inf
>   /
> -inf

Hallo,

Je moet hem wel goed overschrijven he (opgave 34?, grondslagen analyse
II).
Als je hem wel goed overgeschreven hebt kom ik er niet uit.

Anyway I think it's supposed to be the following:

d/dx S_-inf^inf [ sin(xt)*f(t) ] dt = S_-inf^inf [ cos(xt)*t*f(t) ] dt

Let S be the integral sign.
[I hope the translation is correct.]

Fubini says:
Let (omega_i,U_i,mu_i), i=1,2 be two sigma-finit measure-spaces and let
g be an mu_1 x mu_2-
integratable numeric function on omega_1 x omega_2.

Then "some stuff" and S g d(mu_1 x mu_2) = S ( S g(w_1) dmu_1 ) dmu_2 =
S ( S g(w_2) dmu_2 ) dmu_1.
With g(w_1) I mean g_(w_2) : w_1 -> g(w_1,w_2), thus w_2 kept constant.

Since we are talking about function with a Lebesgue-Borel measure we
have:
omega_i = R for i=1,2
U_i = B (the Borel-sets) for i=1,2
mu_i = lamda (the Lebesgue-measure) for i=1,2

We have sin(xt) =  t * S_0^x cos(st) ds and thus
d/dx S_-inf^inf sin(xt)*f(t) dt = d/dx S_-inf^inf [ t * f(t) * S_0^x
cos(st) ds ] dt = (Fubini)
d/dx S_0^x [ S_-inf^inf cos(st)*t*f(t) dt ] ds = S_-inf^inf
cos(xt)*t*f(t) dt

To apply Fubini we must have that the mapping  g: R^2 -> g(R^2) defined
by
g(s,t) = cos(st) * t * f(t) is an lamda^2-measurable numeric function on
R^2.
Since f is a Lebesgue-Borel-measurable function and cos and "t"  are
continuous we have that g
is a Borel-measurable function on R^2.

Exercise: How have we used that S_-inf^inf |t*f(t)| dt < oo ?

Wilbert