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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,

Pete

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