This book explores a new concept of T-shy sets in Radon metric groups which is an extension of the concepts of null sets introduced by J.R. Christensen, J. Mycielski and B R Hunt, T Sauer and J A Yorke for Polish groups. This book considers various interesting applications of the theory of infinite-dimensional cellular matrices for a solution of various initial condition problems. It also includes several direct generalizations of well-known classical results (Lebesgue theorem, Weyl theorem, Rademacher theorem, etc.,) in terms of infinite-dimensional "Lebesgue measures".