In the paper under review the authors are interested in the stability of the class of Riesz sets by union. Let \(G\) be a compact abelian group and \(\Gamma\) be its discrete dual group. A set \(\Lambda\subseteq\Gamma\) is called a Riesz set if every \(\mathcal M_\Lambda(G)=L^1_\Lambda(G)\), where \(\mathcal M(G)\) denotes the space of all regular Borel measures on \(G\), \(L^1(G)\) is its subspace of absolutely continuous measures, and the subscript \(\Lambda\) expresses that we are interested only in those measures \(\mu\) whose Fourier coefficients \(\widehat\mu(\gamma)=\int_G\overline\gamma\,d\mu\) vanish outside \(\lambda\) (\(\overline\gamma\) denotes the character associated with \(\gamma\)). An invariant mean \(M\) on \(L^\infty(G)\) is a continuous linear functional such that \(M(1)=\| M\| =1\) and \(M(f_x)=M(f)\) for every \(f\in L^\infty(G)\) and \(x\in G\) where \(f_x(t)=f(t-x)\). A function \(f\in L^\infty(G)\) is said to have a unique invariant mean if \(M(f)=\widehat f(0)\) for every invariant mean \(M\). A set \(\Lambda\subseteq\Gamma\) is called a Lust-Piquard set (or also totally ergodic) if \(\gamma f\) has a unique invariant mean for every \(f\in L^\infty_\Lambda(G)\) and every \(\gamma\in\Gamma\). The authors prove that the union of a Riesz set and a Lust-Piquard set is a Riesz set. Several results of other authors are consequences of this theorem: The union of a Rosenthal set and a Riesz set is a Riesz set because every Rosenthal set is a Lust-Piquard set; every Lust-Piquard set is a Riesz set; for \(G=\mathbb R/\mathbb Z\), \(\mathbb N\) is not a Lust-Piquard subset of \(\mathbb Z\) since \(\mathbb Z^-\) is a Riesz set and \(\mathbb Z\) is not; and others. Finally, the authors give an example of a Rosenthal subset of \(\mathbb Z\) which is dense in \(\mathbb Z\) for the Bohr topology.