Continuity and singularity of measures under action groups (Q1334368)

The author considers measurable actions of metrizable locally compact groups \(G\) with Haar measure \(m_G\) on measurable spaces \((X, {\mathcal B})\) and the corresponding convolution of \({\mathcal M}(G)\), the positive measures on \(G\), with \({\mathcal M}(X)\). He shows first that a collection of six equivalent conditions on a measure \(\mu\in {\mathcal M}(X)\) established by the author and \textit{H. Sato} in [J. Funct. Anal. 120, 188-200 (1994; Zbl 0812.60036)] can be extended to a list of twelve equivalent conditions, namely: (1) \(\mu\ll m_G* \mu\); (7) \(\mu\) is expressed as \(\rho* \nu^+- \rho* \nu^-\) with \(\rho\in {\mathcal M}(G)\), \(\nu^+\), \(\nu^-\in {\mathcal M}(G)\); (8) \(\mu= f\rho* \nu'\), \(0\leq f\leq 1\rho\in {\mathcal M}(G)\), \(\rho\ll m_G\), \(\nu'\in {\mathcal M}(G)\); (9) \(\mu\ll \nu\) for some \(G\)-quasi-invariant \(\nu\in {\mathcal M}(X)\); (10) \(\mu\ll \nu\) for some \(G\)-invariant \(\nu\in {\mathcal M}(X)\); (11) there exist \(B\in {\mathcal B}(G)\) with \(m_G(B)> 0\) and a measure \(\nu\) on \(X\), such that \(\mu_g\ll \nu\) for all \(g\in B\); (12) \(\mu(A)= 0\) for all negligible sets \(A\). -- Here \(\ll\) means absolutely continuous with respect to. Let \(\mu\perp \nu\) mean that \(\mu\) and \(\nu\) are singular. -- The second result gives a list of six equivalent conditions for \(\mu\in {\mathcal M}(X)\). (1) \(\mu\perp m_G* \mu\); (2) \(\mu_g\perp \mu\), \(m_G\). a.e. \(g\in G\); (3) \(\mu\perp \nu\) for all \(\nu\in {\mathcal M}(X)\) with \(\nu\ll m_G* \nu\); (4) \(\mu\perp\nu\) for all \(G\)-quasi-invariant \(\nu\in {\mathcal M}(X)\); (5) for every \(\nu\in {\mathcal M}(X)\), \(\mu\perp \nu_g\) holds for \(m_G\)- a.e. \(g\in G\); (6) \(\mu(A^c)= 0\) for some negligible set \(A\). -- If \(\lambda\) denotes a \(G\)-quasi-invariant measure on \(X\), then the author proves that \(m_G* \mu\ll \lambda\) if and only if \(\mu\ll \lambda\) and \(m_G* \mu\perp \lambda\) if and only if \(\mu\perp\lambda\).