Duality for real and multivariate exponential families

Consider a measure $\mu$ on ${\mathbb{R}}^n$ generating a natural exponential family $F(\mu )$ with variance function $V_{F(\mu )}(m)$ and Laplace transform $$\exp ({\mathrm{\ell }}_{\mu }(s))={\int }_{{\mathbb{R}}^n}\exp (-\text{<}s,x)\mu (dx)\mathrm{.}$$ A dual measure ${\mu }^{\ast }$ satisfies $-{\mathrm{\ell }}_{{\mu }^{\ast }}^{\mathrm{\prime }}(-{\mathrm{\ell }}_{\mu }^{\mathrm{\prime }}(s))=s\mathrm{.}$ Such a dual measure does not always exist. One important property is $\mathrm{\ell }{\mathrm{"}}_{{\mu }^{\ast }}(m)=(V_{F(\mu )}(m){)}^{-1},$ leading to the notion of duality among exponential families (or rather among the extended notion of T exponential families $TF$ obtained by considering all translations of a given exponential family $F$).