On approximation properties of certain incomplete systems

Let $\{\varphi_n(x)\}$ be a system of almost-everywhere finite measurable functions on $[0,1]$ that has one of the following properties:
I. $\{\varphi_n(x)\}^\infty_{n=1}$ is a system for representing the functions in $L_p[0,1]$, $0<p<1$, by convergent series.
II. $\{\varphi_n(x)\}^\infty_{n=1}$ is a system for representing the functions in $L_p[0,1]$, $0<p<1$, by almost-everywhere convergent series.
III. $\{\varphi_n(x)\}^\infty_{n=1}$ has the strong Luzing $C$-property.
IV. $\{\varphi_n(x)\}^\infty_{n=1}$ can be multiplicatively completed to form a system for representing the functions in $L_p[0,1]$, $p\geqslant1$, by series that converge in the $L_p[0,1]$-metric.
It is shown that if $\{\varphi_n(x)\}^\infty_{n=1}$ is a system having one of the properties I–IV, then any subsystem of it with the form $\{\varphi_k(x)\}^\infty_{k=N+1}$ ($N$ any natural number) also has this property.
Bibliography: 9 titles.