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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

  1. $\mu^{**} = \mu$
  2. 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)

$f^*(w) = \frac{1}{(2 \pi)^{n/2}}\int f(v) e^{- i w(v)} \mathrm{d} \mu(v)$

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  • $\begingroup$ The Fenchet conjugate of a measure is related to entropy and large deviations. You can see that an say, Kallenberg's foundations of probability (Sanov's theorem). $\endgroup$ Sep 20, 2020 at 17:27
  • $\begingroup$ This question is too vague. Perhaps you should describe the context where it appeared, or other properties you would like it to have. This might help people to focus and perhaps find an answer that suits you. $\endgroup$
    – Ruy
    Sep 21, 2020 at 22:12
  • $\begingroup$ @Ruy thanks, posted a follow up math.stackexchange.com/questions/3836435/… $\endgroup$
    – user357269
    Sep 22, 2020 at 20:00
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    $\begingroup$ It is surely a much more appealing question now! I wish I could find the time to try to answer it! $\endgroup$
    – Ruy
    Sep 23, 2020 at 4:27

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