Exponential family

Remarks

The function $A$ is fixed by normalization: $A(\theta) = \log \int h(x) \exp\big(\sum_i \eta_i(\theta) T_i(x)\big) \, \mu(dx)$. The numbers $\eta_i = \eta_i(\theta)$ are the natural parameters, and writing the density in terms of $\eta$ gives the canonical form.

Example: the Bernoulli distribution with parameter $p \in (0, 1)$, with $\mu$ the counting measure on $\set{0, 1}$. Its density is $p^x (1-p)^{1-x} = \exp\big(x \log \tfrac{p}{1-p} + \log(1-p)\big)$. So it is a $1$-parameter exponential family with $h = 1$, $T(x) = x$, $\eta(p) = \log \frac{p}{1-p}$ and $A(p) = -\log(1-p)$.

Non-example: the uniform distributions on $[0, \theta]$ for $\theta > 0$. In an exponential family the set $\set{x : p_\theta(x) > 0} = \set{x : h(x) > 0}$ does not depend on $\theta$, up to a $\mu$-null set, because the exponential factor is positive. For the uniform family this set is $[0, \theta]$, which moves with $\theta$, so no choice of $h$, $T_i$, $\eta_i$, $A$ works.