I'm trying to prove that:
Let $(\mu_k)_{k=1}^{\infty}$ and $(\nu_k)_{k=1}^{\infty}$ be sequences of probability measures on $(\Omega_k, \mathcal{F}_k)$. Consider the product measures on $(\prod_{k=1}^{\infty}\Omega_k, \sigma(\prod_{k=1}^{\infty}\mathcal{F}_k))$: $$\mu=\prod_{k=1}^{\infty} \mu_k, \mbox{ }\mbox{ }\mbox{ }\nu=\prod_{k=1}^{\infty} \nu_k$$ Prove that $$H(\mu,\nu)=\prod_{k=1}^{\infty} H(\mu_k,\nu_k).$$
Hellinger integral is defined by $$H(\mu,\nu)=\int_{\Omega} \sqrt{\dfrac{d\mu}{d\zeta}\dfrac{d\nu}{d\zeta}} d\zeta$$ where $\mu$ and $\nu$ are absolutely continuous with the respect to the probbaility measure$\zeta=\frac{1}{2}(\mu+\nu)$ on $(\Omega, \mathcal{F})$
Does anybody know how to do it?