Given $X,\theta\in\mathbb{R}^d$, the density of a natural exponential family is defined as
$f(x|\theta)=h(x)e^{\theta\cdot x-\psi(\theta)}$
with the natural parameter space
$\Theta=\left\{\theta:\int e^{\theta x}h(x)\;dx<\infty\right\}$.
The textbook I am reading says that the family
$\pi(\theta|\mu,\lambda)= K(\mu,\lambda)e^{\theta \mu-\lambda\psi(\theta)},\quad\lambda>0,\frac{\mu}{\lambda}\in\Theta^\circ$
is a conjugate prior family ($\Theta^\circ$ denotes the interior of $\Theta$). Calculating this out is straightforward, however I cant figure out how to show that the given density is in fact a density, ie.
$\int \pi(\theta|\mu,\lambda)\;d\theta<\infty$
I have tried for a long time, and looked at the book but I cant even find a hint at what to do. I would appreciate any help.
EDIT: I understand that $1/K(\mu,\lambda)=\int e^{\theta\mu-\lambda\psi(\theta)}\;d\theta$
My problem is proving that this integral exists.
