Let $(P_\theta)_{\theta\in\Theta}$ be a family of probability distributions on some measurable space $\mathcal X$, where $\Theta$ is some open real interval. Let $\nu$ be a $\sigma$-finite measure on $\mathcal X$ that dominates all the $P_{\theta}$'s (let us assume that such a $\nu$ exists). For each $\theta\in\Theta$, let $f(\cdot,\theta)$ be a density of $P_\theta$ with respect to $\nu$, and assume that for $\nu$-almost all $x\in\mathcal X$, $f(x,\cdot)$ is differentiable (as a function of $\theta$).
Now, let $\phi$ be a measurable function from $\mathcal X$ to some other measurable space $\mathcal Y$. Denote by $\mu$ the push-forward of $\nu$ by $\phi$, and by $Q_\theta$ the push-forward of $P_\theta$ by $\phi$, for all $\theta\in\Theta$.
Is it true that for all $\theta$, $Q_\theta$ has a density $g(\cdot,\theta)$ with respect to $\mu$ such that $g(y,\cdot)$ is differentiable for $\mu$-almost all $y\in\mathcal Y$?
The reason why I'm asking is that I would like to prove rigorously that if a parametric model has a Fisher information, then the push-forward model also has a well-defined Fisher information.
Thanks!