Equivalence of Nikodým's theorems for group measures (Q1096032)

For any topological group G two notions of boundedness may be considered: \(A\subseteq G\) is 1-bounded (resp. 2-bounded) if for each neighbourhood V of zero in G there is \(n\in N\) (and \(g_ 1,...,g_ n\) in G) such that \(A\subseteq \oplus^{n}V\) (resp. \(A\subseteq \cup^{n}_{k=1}(g_ k+\oplus^{n}V)).\) We say a ring \({\mathcal R}\) of subsets has Nikodým property for i-boundedness (abbr. \({\mathcal R}\in {\mathcal N}_{loc}(i))\) provided for any Abelian topological group G, any neighbourhood V of a zero in G and any family M of V-exhausted measures \(m:{\mathcal R}\to G\) if \(\{m(x):m\in M\}\) is i-bounded for any \(x\in {\mathcal R},\) then \(\{\) m(x):m\(\in M\), \(x\in {\mathcal R}\}\) is i-bounded \((i=1,2)\). The following theorem is proved in the paper: \[ {\mathcal R}\in {\mathcal N}_{loc}(1)\quad iff\quad {\mathcal R}\in {\mathcal N}_{\log}(2). \] In the proof at first topology is omitted by reducing to measures with finitely many values and then using a little ``group theory'' one has to prove only the case G is the group of roots of unity in the field of complex numbers. Other extensions of Nikodým's theorem to group-valued measures one can find in \textit{E. Guariglia} [Matematiche 37(1986), 328-342 (1982; Zbl 0598.28016)] and \textit{C. Constantinescu} [Libertas Math. 1, 51-73 (1981; Zbl 0482.28009)].