In the answer to a previous question of mine, an anonymous user invoked the Layer Cake Representation in order to reduce proving \begin{equation} \Re\left[ \int_{-\infty}^\infty \frac{1+iy}{1+i(y-t)} dG(t)\right] \ge \Re\left[\int_{-\infty}^\infty \frac{1}{1-it} dG(t) \right] \end{equation} for all $y>0$, to proving \begin{equation} \Re\left[ \int_{-a}^a \frac{1+iy}{1+i(y-t)} dt\right] \ge \Re\left[\int_{-a}^a \frac{1}{1-it} dt \right] \end{equation} for all $y,a>0,$ where $dG$ is the meaure of a symmetric probability distribution (non-increasing on $0 <t<\infty$).

I have Lieb and Loss's analysis text and I think I understand the theorem and the corresponding proof (Theorem 1.13, for reference).

My questions are:

  1. How does one actually use the Layer Cake representation to evaluate integrals over different measures and are there restrictions on measures that make evaluating such integrals easier/harder?

  2. How was it applied to in the problem above?

Many thanks,



Your Answer

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

Browse other questions tagged or ask your own question.