Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$.

Is the following true?

The span of the shifts of $\mu$ by vectors in $\ell^2$ is dense in total variation in the space of $\gamma$-absolutely continuous measures iff

 

$$\forall c>0: \inf_{\Vert \xi \Vert_{\ell^2} \le c} \left| \intop e^{i \langle x, \xi \rangle} \mu(dx) \right| > 0$$

Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$.

Is the following true?

The span of the shifts of $\mu$ by vectors in $\ell^2$ is dense in total variation in the space of $\gamma$-absolutely continuous measures iff

$$\forall c>0: \inf_{\Vert \xi \Vert_{\ell^2} \le c} \left| \intop e^{i \langle x, \xi \rangle} \mu(dx) \right| > 0$$