I'm trying to derive the general process of changing variables so that an exponential family has zero carrier measure. Distributions in the exponential family have cdf
$dF(\mathbf{x}|\boldsymbol\eta) = \exp\left({\boldsymbol\eta \cdot T(\mathbf{x}) - g(\boldsymbol\eta)}\right)\, dh(\mathbf{x}).$
I guess this is a Lebesgue integral, but I don't understand the notation that well.
I would like to find the function $z$ so that
$dF(z(\mathbf{y})|\boldsymbol\eta) = \exp\left({\boldsymbol\eta \cdot T(z(\mathbf{y})) - g(\boldsymbol\eta)}\right)\, dh(z(\mathbf{y})).$
and I want the function $h$ to disappear. I want the it to be the standard Lebesgue measure.
So, for the poisson distribution, $h(x) = \frac{1}{x!}$. What is $z$? Is it just $h^{-1}$?