Analysis Problem of the Day 101

Today’s problem appeared as Problem 1 on the UCLA Fall 2019 Analysis Qual:

Problem 1. Given $\sigma$ -finite measures $\mu_1 \ll \nu_1, \mu_2 \ll \nu_2$ on a measure space $(X,\mathcal{X}),$ prove that the product measures $\mu_1 \otimes \mu_2$ and $\nu_1 \otimes \nu_2$ on $(X \times X, \mathcal{X} \otimes \mathcal{X})$ satisfy $\mu_1 \otimes \mu_2 \ll \nu_1 \otimes \nu_2$ and

$\frac{d(\mu_1 \otimes \mu_2)}{d(\nu_1 \otimes \nu_2)} (x,y)= \frac{d\mu_1}{d\nu_1} (x) \frac{d\mu_2}{d\nu_2} (y)$

in the sense of Radon-Nikodym derivatives $\nu_1 \otimes \nu_2$ -a.e.

Solution: It suffices to show the representation formula, since it would imply the absolute continuity of the product measures. Since $\mu_1 \ll \nu_1, \mu_2 \ll \nu_2,$ the Radon-Nikodym derivatives $\frac{d\mu_1}{d\nu_1}, \frac{d\mu_2}{d\nu_2}$ are well-defined and

$\int \chi_A(x) \chi_B(y)d\mu_1(x)d\mu_2(y) = (\mu_1 \otimes \mu_2)(A \times B) = \mu_1(A)\mu_2(B) =$

$\int \chi_A (x)\frac{d\mu_1}{d\nu_1} (x)d\nu_1(x) \int \chi_B (y) \frac{d\mu_2}{d\nu_2}(y) d\nu_2(y)$

$= \int \chi_A(x)\chi_B(y) \frac{d\mu_1}{d\nu_1}(x)\frac{d\mu_2}{d\nu_2}(y)d\nu_1(x)d\nu_2(y),$

which implies that the representation formula holds valid on rectangles of the form $A \times B.$ We know by the Caratheodory construction that any $\nu_1 \otimes \nu_2$ -measurable set $C$ may be approximated in measure by a countable union of such rectangles. Since characteristic functions of finite unions and intersections of rectangles can be written as a finite disjoint sum of rectangles, it follows by linearity that we can approximate the characteristic function of an arbitrary measurable set by a linear combination of characteristic functions of rectangles. Finally, since the representation formula holds for rectangles, by a density argument we extend the formula to hold for characteristic functions of measurable sets. But this precisely implies that $(\mu_1 \otimes \mu_2)(C) = \int_C \frac{d\mu_1}{d\nu_1}(x) \frac{d\mu_2}{d\nu_2}(y) d\nu_1(x) d\nu_2(y),$ so $\mu_1 \otimes \mu_2 \ll \nu_1 \otimes \nu_2$ and the desired equality holds $\nu_1 \otimes \nu_2$ -a.e.

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