Let $V$ be a finite dimensional real vector space with a (finite) measure $\mu$ on it.
Can one define $\mu^*$ on the dual space $V^*$ such that
- $\mu^{**} = \mu$
- The construction doesn't depend on a choice of basis
I realize this may not uniquely characterize what $\mu^*$ does, but is there a common construction?
I know there are ways to do this for functions.
If $f: V \to \mathbb{R}$ is convex (+ conditions), there's the Fenchel conjugate:
$$f^*(w) = \sup_v \{w(v) - f(v)\} $$
Or if we take $\mu$ to be the Lebesgue measure and $f$ is $L^1$ (+ conditions)
$f^*(w) = \frac{1}{(2 \pi)^{n/2}}\int f(v) e^{- i w(v)} \mathrm{d} \mu(v)$