This fact appears in my statistics textbook (Pg 543, statistical decision theory and bayesian analysis). it says : for normal distribution the generalized bayes estimator becomes \begin{align*} \delta_F(x)&=\nabla \log[(2\pi)^{-p/2}e^{-|x|^2/2}\int e^{x'\theta}e^{-|\theta|^2/2}d(F\theta)]-\nabla \log[e^{-|x|^2/2}]\\ &=\nabla \log \int e^{x'\theta}e^{-|\theta|^2/2}F(d\theta) \end{align*} This integral is a Laplace transform of $exp\{-\frac{1}{2}|\theta|^2\}dF(\theta)$, which is known to be infinitely differentiable.
It seems like this is a well-known fact. However, I don't know how it was deduced since I don't know about laplace transform. I want to know why this is indeed the Laplace transform of $exp\{-\frac{1}{2}|\theta|^2\}dF(\theta)$ since the $log$ doesn't make much sense at first look. I also want to know why this is infinitely differentiable.
Thanks in advance for any help.