On Bernstein's theorem about a sequence of best approximations in spaces $L^{\varphi}$.

Let $T=(T,\Sigma,\mu)$ be a measure space, $\sigma$-algebra $\Sigma$ containing all the sets of measure zero and a set $E$ with $0<\mu(E)<\infty$; let $Y$ be an $F$-space with a quasinorm $|\cdot|_1$ nondecreasing along each ray emanating from the origin, $\varphi\colon[0,\infty)\to[0,\infty)$ be a continuous nondecreasing semiadditive function, $\varphi(\alpha)=0\Leftrightarrow\alpha=0$. Denote by $L^{\varphi}=L^{\varphi}(T,Y)$ the linear space of all measurable mappings $f\colon T\to Y$ with $|f|:=\int_T\varphi(|f(t)|_1)d\mu(t)<\infty$. Let $L_n$ be asequence of finite-dimensional subspaces of $L^{\varphi}$ such that $L_n\subset L_{n+1}$, $L_n\neq L_{n+1}$. The problem of existence of an element $f\in L^{\varphi}$ with the preassigned best approximations $a_n$ – distances from $f$ to $L_n$ – is considered.