On the distributions of logarithmic derivative of differentiable measures on \({\mathbb{R}}\) (Q915250)

It is proved for an arbitrary probability distribution P with mean 0 (P is not the Dirac measure), that there exists a differentiable probability measure \(d\mu (x)=f(x)dx\), such that \[ P(E)=\mu (x | \quad f'(x)/f(x)\in E) \] for all E from the usual Borel field on \({\mathbb{R}}\). The function \(f_ k(x)=f(x+k)\) also satisfies \(\mu_{f_ k}=P\) (k constant). If \(P(E)=P(-E)\), then f is an even function and \(f(0)>0\).