A note to a theorem of Talaljan (Q1307367)

The aim of this paper is to prove the Theorem. Let \((X,\mu)\) be a measure space, let \(\mu\) be nonatomic, and let \(\{\phi_n\}^\infty_1\) be a complete orthonormal system in \(L^2_\mu(X)\). Then for every \(\{c_n\}^\infty_1 \in C_0\), except a set of first category, the partial sums \[ \left\{\sum^N_{n=1} c_n\phi_n:N=1,2,\dots\right\} \] (a) in case \(\mu(X)<\infty\): form a dense set in \(L^p_\mu(X)\) for every \(p\in[0,1)\); (b) in case \(\mu(X)=\infty\), \(Y\subseteqq X\) is an arbitrary measurable set for which \ \ \ \ \ \ (i)\ \ \(\mu(Y)=\infty\): form a pointwise dense set in \(L^p_\mu(Y)\) for every \(p\in[0,1)\); \ \ \ \ \ \ (ii)\ \(\mu(Y)<\infty\): form a locally dense set in \(L^p_\mu(Y)\) for every \(p\in[0,1)\).