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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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How do you define the extension to the entire product $\sigma$-algebra when following your idea? Do you know how to extend a signed (or complex) measure from an algebra to a $\sigma$-algebra? –  t.b. Jan 12 '12 at 8:55
    
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. –  Mark Jan 12 '12 at 10:59
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