Orateur
Description
The questions raised here grew from the desire to give an integral representation for members of the dual of $BV$, the Banach space of functions of bounded variation. This potentially has application to the calculus of variations since $BV$ dual contains subgradients of energy functionals to be minimized. The question quickly links to that of identifying the dual of $L^1(\mathbb{R}^m,\mathcal{A},\mathcal{H}^{m-1})$ where $\mathcal{H}^{m-1}$ is the Hausdorff measure. Whether the corresponding canonical map $\Upsilon : L^\infty \to (L^1)^*$ is injective or not depends upon the $\sigma$-algebra $\mathcal{A}$. For $\mathcal{A}$ being the $\sigma$-algebra of measurable sets in the sense of Caratheodory, the surjectivity of $\Upsilon$ is undecidable in ZFC. This calls for trying to associate with every measure space $(X,\Sigma,\mu)$, in a universal way, a new measure space $(\hat{X},\hat{\Sigma},\hat{mu})$ with respect to which the Radon-Nikodym theorem holds -- alternatively such that the corresponding $\hat{\Upsilon}$ is an isometric isomorphism -- and $L^1(X,\Sigma,\mu) \cong L^1(\hat{X},\hat{\Sigma},\hat{mu})$. I will explain how this is better stated in a specific category whose objects are "measurable spaces with negligibles". In that context, the existence of the universal "Radon-Nikodymification" is obtained via several applications of Zorn's Lemma and, therefore, is not much of practical use in general. In a particular case that pertains to $BV$ dual, specifically when $\mu$ is an integral geometric measure (instead of Hausdorff measure), I will show that $\hat{X}$ can be described explicitly as a fibered space of $\mathcal{R}^m$ whose fiber above $x$ consists of germs of rectifiable sets through $x$. Part of these results have been obtained in collaboration with Philippe Bouafia.
