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Given a locally compact abelian group $G$ with Haar measure $m$, it is well-known that there exist measures $\mu\in M(G)$ which are singular (with respect to $m$), but the convolution product $\mu\ast\mu$ is absolutely continuous.

(Q.1) Is there any condition characterizing the measures $\mu$ which satisfy the following (opposite) $tauberian$ property: given $\nu \in M(G)$, $\mu\ast\nu$ absolutely continuous implies $\nu$ absolutely continuous?

If $\mu_1$ is invertible in $M(G)$ and $\mu_2$ is absolutely continuous, then $\mu=\mu_1+\mu_2$ satisfies this tauberian property.

(Q.2) Are there further examples?

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  • $\begingroup$ What is the definition of invertible measure? and what do you mean by $\mu_{2}$ to be a.c? wrt to $m$? $\endgroup$
    – Asaf
    Commented Sep 16, 2013 at 13:14
  • $\begingroup$ $(M(G),*)$ is a commutative Banach algebra. $\mu$ invertible means that there is $\nu$ such that $\mu * \nu =\delta_e$, where $e$ is the unit of $G$ ($\delta_e$ is the unit of $M(G)$), and $\mu_2$ a.c. wrt m means $A$ Borel subset of $G$ and $m(A) =0$ implies $\mu_1(A)=0$; equiv., there is $f_1 \in L_1(m)$ such that $d\mu_1 = f_1\cdot d m$. $\endgroup$
    – M.González
    Commented Sep 16, 2013 at 14:49
  • $\begingroup$ Every $\mu\in M(G)$ can be decomposed as $\mu= \mu_d + \mu_c$ where $\mu_d$ is the discrete (atomic) part and $\mu_c$ is the continuous part (no atoms).\newline $\endgroup$
    – M.González
    Commented Sep 17, 2013 at 7:14
  • $\begingroup$ CONTINUATION: It was proved by R. Doss [Studia Math. 45 (973), 111-117] that for every continuous measure $\mu$ there exists a non absolutely continuous measure $\mu$ such that $\mu *\nu$ is absolutely continuous. Therefore a measure satisfying the tauberian property has non-zero atomic part. $\endgroup$
    – M.González
    Commented Sep 17, 2013 at 7:21
  • $\begingroup$ @M. Gonzalez, the Lebesgue decomposition statement you've written is false, as $\mu$ can be decomposed as a.c. part plus a singular part, but the singular need not be atomic. It can be for example a Cantor measure, or any measure of dimension less than $m$, but need not be discrete. $\endgroup$
    – Asaf
    Commented Sep 19, 2013 at 21:03

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