# References for formal powers of measures

In Information geometry, Ay et al. define the space of formal $$r$$th powers* of signed measures as the limit $$\mathcal S^r(\Omega) = \injlim L^{1/r}(\Omega, \mu)$$ of maps $$\phi\mapsto (\mu/\nu)^r\phi :$$ $$L^{1/r}(\Omega,\mu)\to L^{1/r}(\Omega,\nu)$$, $$\mu\leq\nu$$ (referencing only an exercise in Neveu’s Bases mathématiques du calcul de probabilités). It appears to me that this is essentially the same thing as Dmitri Pavlov’s definition (in arχiv:1309.7856 §5.10 or on MO) of a measure-independent space $$\mathrm L_r(\Omega)$$ as a limit of measure-dependent spaces $$\mathrm L_r(\Omega,\mu)\equiv L^{1/r}(\Omega,\mu)$$. It also (I think) reproduces the standard notion of $$r$$-densitites on manifolds.

Overall, this seems like a pretty elementary and even essential construction, but all the references above only mention it in passing and with scant references to the literature. I’d like to find an exposition that treats it with the attention it deserves, perhaps alongside or shortly after the usual classes of measures and measurable functions.

* The book also sometimes refers to them as formal $$r$$th roots (with the same $$r$$!)—mistakenly, I think.