Categorial properties of the set of nonuniqueness points in the space \(L_ \phi\) (Q1813390)

The paper is concerned with certain metric linear spaces \(L_ \varphi\) of measurable functions on a measure space \((T,\Omega,\mu)\); a space \(L_ \varphi\) being defined by a function \(\varphi\) satisfying specified conditions and its metric being defined by \(\rho(f,g)=\int_ T\varphi(f- g)d\mu\). For these spaces the paper presents the following theorem: If \(\mu\) is a non-atomic measure and \(M\subseteq L_ \varphi\) then the set of \(f\in L_ \varphi\) such that \(f\) is both a point of approximative compactness and a point of non-uniqueness for best approximation from \(M\) is a set of first category in \(L_ \varphi\). The theorem generalises a result of \textit{S. Ya. Khavinson} and \textit{Z. S. Romanova} [Mat. Sb. n. Ser. 89 (131), 3-15 (1972; Zbl 0252.41037)] for linear subspaces of \(L_ 1\) spaces.