# Application of 'Total variation of a complex measure is finite.'

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.

## 1 Answer

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.