1
$\begingroup$

I have a 'Real and Complex analysis' by Rudin. And theorem 6.4 says 'Total variation of a complex measure is finite.'

Q: What is an application of this property?

I've read all the exercise in the book and I couldn't find any application of the theorem. Thanks.

$\endgroup$
0
$\begingroup$

I don't know of any exercises that make use of this result, but I can tell you of a point in the text where he makes crucial use of it: the proof of the theorem of the Brothers Riesz (17.13).

Briefly, to show that the measure is absolutely continuous (with respect to normalized Lebesgue measure on $\mathbb T$), he obtains a Radon-Nikodym derivative that is in $L^1(\mathbb T)$ as it is the boundary values of a function in $H^1(\mathbb D)$. Showing that the function is in $H^1(\mathbb D)$ rests upon being able to show that it is analytic, which comes from the main hypothesis of the theorem, and that its $H^1$ norm is finite, which comes from the fact that it is bounded above by $\lvert \mu \rvert$ and $\mu$ is a complex Borel measure.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.