Suppose $(X, \mathcal{F})$ is a measurable space and $\left\{F_\theta, \theta \in \Theta\right\}$ is a distribution family on $(X, \mathcal{F})$. When $\left\{F_\theta, \theta \in \Theta\right\}$ is dominated by a $\sigma$ finite measure $\mu,\left\{F_\theta, \theta \in \Theta\right\}$ is a exponential family if and only if $$ \frac{d F_\theta}{d \mu}=\exp \left\{\eta(\theta)^{\top} T(x)-\xi(\theta)\right\} h(x), \forall \theta \in \Theta, $$ where $h(x)$ is non-negative measurable which is stated in many references like Lehman and Casella, Theory of Estimation.
Then is it true that $\frac{\partial \ln (F_\theta)}{\partial \theta}$ exists? or under what condition it does exists?
Where $F_\theta$ is seen as distribution function of exponential family i.e $$ F_\theta(B) = \int_B \exp \left\{\eta(\theta)^{\top} T(x)-\xi(\theta)\right\} h(x) \, d \mu(x), \quad B \in \mathcal{B}, $$ where $\mathcal{B}$ is the sigma-algebra that is the domain of $\mu$.
