Complex Measures: Integration

Question

Reference

A rigorous treatment can be found in: Complex Measures: Integrability

Problem

Integration w.r.t. complex measure usually is defined via the Radon-Nikodym derivative: $$\int f\mathrm{d}\mu:=\int fu\mathrm{d}|\mu|\quad(\mathrm{d}\mu=u\mathrm{d}|\mu|)$$ Alternatively one could define: $$\int f\mathrm{d}\mu:=\sum_{\alpha=0\ldots3}i^\alpha\int f\mathrm{d}\mu_\alpha$$ with the decomposition: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$

How to check that these definitions agree formally?

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Answer 1

Formally it holds: $$\int fu\mathrm{d}|\mu|=\sum_{\alpha}i^\alpha\int f_\alpha u\mathrm{d}|\mu|=\sum_{\alpha,\beta}i^{\alpha+\beta}\int f_\alpha u_\beta \mathrm{d}|\mu|=\sum_{\beta}i^\beta\int f\mathrm{d}\mu_\beta$$