TITLE: Radon-Nikodym derivative of product measure
QUESTION [16 upvotes]: For $j=1,2$, let $\nu_{j},\mu_{j}$ be $\sigma$-finite measures on $(X_{j},\mathcal{M}_{j})$ such that $\nu_{j}\ll\mu_{j}$. I want to show that $\nu_{1}\times\nu_{2}\ll\mu_{1}\times\mu_{2}$ and that $\frac{d(\nu_{1}\times\nu_{2})}{d(\mu_{1}\times\mu_{2})}(x_{1},x_{2})=\frac{d\nu_{1}}{d\mu_{1}}(x_{1})\frac{d\nu_{2}}{d\mu_{2}}(x_{2})$.
I tried to show that if $E\in\mathcal{M}_{1}\otimes\mathcal{M}_{2}$ and $\mu_{1}\times\mu_{2}(E)=0$, then $\nu_{1}\times\nu_{2}(E)=0$. This criterion holds for measurable rectangles so then I tried considering $\{E\in\mathcal{M}_{1}\otimes\mathcal{M}_{2}|\mu_{1}\times\mu_{2}(E)>0\;\text{or}\;\nu_{1}\times\nu_{2}(E)=0\}$ and show that it is a $\sigma$-algebra containing all measurable rectangles but I guess it didn't quite work.

REPLY [10 votes]: Denote $X=X_1\times X_2$, $\mu=\mu_1\times\mu_2$, $\nu=\nu_1\times\nu_2$, $\mathcal{M}=\mathcal{M}_1\times\mathcal{M}_2$ and 
$$f(x_1,x_2)=\dfrac{d\nu_1}{d\mu_1}(x_1)\dfrac{d\nu_2}{d\mu_2}(x_2).$$ 
Evidently $f$ is non-negative,  $\mathcal{M}$ measurable and $\mu$ locally integrable. Then as @copper.hat commented, by Fubini's theorem(or Tonelli's theorem instead), we have: 
$$\nu(E_1\times E_2)=\nu_1(E_1)\cdot\nu_2(E_2)=\int_{E_1}\dfrac{d\nu_1}{d\mu_1}d\mu_1\times\int_{E_2}\dfrac{d\nu_2}{d\mu_2}d\mu_2=\int_{E_1\times E_2}f~d\mu.$$
Consider the collection 
$$\mathcal{C}:=\{E\in \mathcal{M}:\nu(E)=\int_E f~d\mu\}$$ 
and we want to show $\mathcal{C}=\mathcal{M}$. We have shown $\Pi:=\{E\in\mathcal{M}:E=E_1\times E_2\}\subset \mathcal{C}$, it suffices to show $\mathcal{C}$ is a $\sigma$-algebra. First let us assume that $\mu(X)<\infty$. Since $\Pi$ is a $\pi$-system, we only need to show $\mathcal{C}$ is a $\lambda$-system: (i) $X\in\mathcal{C}$; (ii) if $D,E\in\mathcal{C}$ with $D\subset E$, then $E\setminus D\in\mathcal{C}$; and  if $\{E_n:n\ge 1\}\subset \mathcal{C}$ is an increasing sequence, then $\cup_n E_n\in\mathcal{C}$.  The first two items are obvious and the last one follows from monotone convergence theorem. Now we have proved $\mathcal{C}=\mathcal{M}$ with the additional assumption $\mu(X)<\infty$. For general $X$, let $\{F_n=F_{1,n}\times F_{2,n}:n\ge 1\}\subset\Pi\subset\mathcal{C}$ be such that $\mu(F_n)<\infty$ for every $n$ and $\cup_n F_n=X$. Then the argument before implies that for every $n$, $\{E\in \mathcal{C}:E\subset F_n\}=\{E\in \mathcal{M}:E\subset F_n\}$. Then given $E\in\mathcal{M}$, $E\subset F_n\in\mathcal{C}$ for every $n$, so again by monotone convergence theorem, $E\in\mathcal{C}$, which completes the proof.
Therefore, when $E\in\mathcal{M}$, $\nu(E)=\int_E f~d\mu$. The conclusion follows.