Geodesic
Systems of functions which are complete in measure
Matematičeskie zametki, Tome 18 (1975) no. 6, pp. 815-824.

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In this note it is proved that if a complete orthonormal system $\{\varphi_n\}$ in $L_2[0,1]$ contains a subsystem $\{\varphi_{n_k}\}$ of a lacunary order $p>2$, then for some bounded measurable function $h(x)$ the system $\{h(x)\varphi_n(x)\}_{n\ne n_k}$ is complete in $L_2[0,1]$. 
@article{MZM_1975_18_6_a2,
     author = {N. B. Pogosyan},
     title = {Systems of functions which are complete in measure},
     journal = {Matemati\v{c}eskie zametki},
     pages = {815--824},
     publisher = {mathdoc},
     volume = {18},
     number = {6},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_18_6_a2/}
}
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N. B. Pogosyan. Systems of functions which are complete in measure. Matematičeskie zametki, Tome 18 (1975) no. 6, pp. 815-824. http://geodesic.mathdoc.fr/item/MZM_1975_18_6_a2/