The paper studies explicit constructions of expander families, the Cayley graphs determined by a group and a generator set. These constructions are far from being trivial, see, for example, \textit{A. Lubotzky, R. Phillips} and \textit{P. Sarnak} [Combinatorica 8, No. 3, 261-277 (1988; Zbl 0661.05035)] and \textit{G. A. Margulis} [Probl. Inf. Transm. 24, No. 1, 39-46 (1988; Zbl 0708.05030)]. The author proves results concerning these constructions. Perhaps the most interesting one is a connection, which characterizes the expanding families by measure-theoretic properties of a corresponding group. Namely, the family is constructed by factoring a \(\Gamma\) (infinite) group, such that \(F\) generates \(\Gamma\), and \(N_i\) is a sequence of finite index normal subgroups of \(\Gamma\). It is shown that the corresponding Cayley graphs form an expanding family iff the \(\mu\)-integration is the unique \(\Gamma\)-invariant mean on \(L^\infty(G,\mu)\), where \(G\) is the inverse limit of the sequence \(\Gamma/N_i\).