# Fractional Radon - Nikodym derivative

Given $f$ a function measurable in $[0,\infty]$ defined as $$\frac{d\mu}{d\nu}$$

such that, for every measurable set $A$ we have:

$$\mu(A)=\int_A{}fd\nu$$ the function $f$ is called $Radon - Nikodym\ derivative$ of $\mu$ respect to $\nu$

My question is: is it possible to define a $fractional\ Radon - Nikodym\ derivative$ of $\mu$ respect to $\nu$?

Thanks.

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Fractional derivatives depend upon the functional calculus of the derivative. You have to check, on which vectorspace the Radon Nikodym derivative should operate, give the vector space a pre Hilbert space structure and that the Radon Nikodym derivate should be a normal operator. I can just come up with trivial constructions. Perhaps you should include the aim of such a construct.... –  Marc Palm Feb 21 '12 at 12:32
I agree. You will need to say what properties you want this "fractional derivative" to have. –  Gerald Edgar Feb 21 '12 at 15:11
$\mu$ and $\nu$ are equivalent gaussian measures –  Riccardo.Alestra Feb 21 '12 at 17:37