Given some measurable space $\left(X,\mathcal{F}\right)$ and two probability measures $\mu$ and $\nu$ on this space one can define $$H_{\theta}(\mu,\nu)=\int\left(\frac{d\mu}{d\lambda}\right)^{\theta}\left(\frac{d\nu}{d\lambda}\right)^{1-\theta}d\lambda $$ where $\theta\in[0,1] $ and $\lambda$ is any positive measure such that $\mu,\nu<<\lambda$ (it is in fact independent of the choice of this measure, and we can in fact assume it is also a probability measure).
Now let $\mathbb{X}$ be a polish space, $\mathcal{F}=\mathbb{B}(\mathbb{X}) $, and consider the countable product of measurable spaces $\left(\prod_{n}\mathbb{X},\otimes_{n}\mathcal{F}\right) $ , and the following measures $ \mu=\prod_{n}\mu_{n},\:\nu=\prod_{n}\nu_{n} $, on this space. Where $\mu_{n}$ and $\nu_{n}$ are probability measures on $\left(\mathbb{X},\mathcal{F}\right) $.
I wish to show that $H_{\theta}(\mu,\nu)=\prod_{n=1}^{\infty}H_{\theta}(\mu_{n},\nu_{n}).$
It is trivial for the case $\theta=0,1$, so assume $\theta\in(0,1)$.
I can show that $H_{\theta}(\mu_{1}\times\mu_{2},\nu_{1}\times\nu_{2})=H_{\theta}(\mu_{1},\nu_{1})H_{\theta}(\mu_{2},\nu_{2})$
From which it follows that, $$H_{\theta}(\mu,\nu)=\prod_{n=1}^{N}H_{\theta}(\mu_{n},\nu_{n})H_{\theta}(\prod_{n=N+1}^{\infty}\mu_{n},\prod_{n=N+1}^{\infty}\nu_{n}). $$ And since (by Hölder's inequality), $$H_{\theta}(\prod_{n=N+1}^{\infty}\mu_{n},\prod_{n=N+1}^{\infty}\nu_{n})\leq1,$$ it only remains to show that $$H_{\theta}(\prod_{n=N+1}^{\infty}\mu_{n},\prod_{n=N+1}^{\infty}\nu_{n})\geq1. $$ Since then $$H_{\theta}(\mu,\nu)=\lim_{N}\prod_{n=1}^{N}H_{\theta}(\mu_{n},\nu_{n})\lim_{N}H_{\theta}(\prod_{n=N+1}^{\infty}\mu_{n},\prod_{n=N+1}^{\infty}\nu_{n})=\prod_{n=1}^{\infty}H_{\theta}(\mu_{n},\nu_{n}). $$ I am kind of stuck here. Maybe this idea is going nowhere. I am not very familiar with infinite product measures and their integrals. Any pointers would be appreciated.