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Given $\mu, \nu$ complex Borel measures on $\mathbb{R}^n$, then product measure $\mu \times \nu$ on $\mathbb{R}^n \times \mathbb{R}^n$ is defined by

$d(\mu \times \nu)(x, y) = \frac{d\mu}{d|\mu|} (x) \frac{d\nu}{d|\nu|} (y) d(|\mu|\times |\nu|)(x, y)$

Why do we need to define product measures in such a way, and can anyone give some insight as to why this definition is natural? What is wrong with mimicking the usual way of defining product measures for (non-complex) measures? (i.e. start with a set $E$ that is the finite disjoint union of rectangles $A_i \times B_i$ and set $(\mu \times \nu)(E) = \sum \mu(A_j) \nu(B_j)$.

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    I guess this can be done using the Hahn-Jordan decomposition (writing a complex measure as a complex-linear combination of four positive measures), but if I remember correctly, at least one proof of the Hahn-Jordan decomposition theorem uses the Radon-Nikodym theorem.2012-01-12

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