Exponential families and full rank

Let $(\mathcal{X},\mathcal{A})$ be standard Borel, let $\mu$ be a $\sigma$-finite positive measure, let $h:\mathcal{X}\to [0,\mathrm{\infty })$ be measurable, and let $T:\mathcal{X}\to {\mathbb{R}}^k$ be a measurable statistic. For $\eta \in {\mathbb{R}}^k$ put

$Z(\eta )=\int h(x)e^{\eta \cdot T(x)}\mu (dx),H=\{\eta :0<Z(\eta )<\mathrm{\infty }\},A(\eta )=\log Z(\eta )\mathrm{.}$

For a parameter set $\mathrm{\Omega }\subseteq H$, the associated exponential family has laws $P_{\eta }$ with density

$p_{\eta }(x)=h(x)\exp \{\eta \cdot T(x)-A(\eta )\}(\eta \in \mathrm{\Omega })\mathrm{.}$

The normalization $A(\eta )=\log Z(\eta )$ makes each density integrate to one. In this item full-rank means that $\mathrm{\Omega }$ contains a nonempty open subset of ${\mathbb{R}}^k$ and there do not exist $b\mathrm{\ne }0$ and $c\in \mathbb{R}$ with $b\cdot T=c$ $\mu$-almost everywhere on $\{h>0\}$. A parameter set containing an open subset is also called regular in some sources; terminology varies, so this definition controls here. The condition is stronger than needed for the completeness theorem below but rules out redundant displayed coordinates.