A measure (u) on (left(mathbb{R}^{n}, mathscr{B}left(mathbb{R}^{n}ight)ight)) is called quasi-invariant if for all (N in mathscr{B}left(mathbb{R}^{n}ight)) with (u(N)=0)

Question:

A measure $u$ on $\left(\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}ight)ight)$ is called quasi-invariant if for all $N \in \mathscr{B}\left(\mathbb{R}^{n}ight)$ with $u(N)=0$ it holds that $u(N+x)=0$ for all $x \in \mathbb{R}^{n}$. Show that $u \ll \lambda$ and $\lambda \ll u$, where $\lambda$ is $n$-dimensional Lebesgue measure.

[ consider the convolution $u \star \lambda(N)$.]

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