Geodesic
Rearrangements of series in $L_p$
Matematičeskie zametki, Tome 14 (1973) no. 1, pp. 31-38.

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In this note we show that the condition $$\sqrt{\sum_{n=1}^\infty f_n^2(x)}\in L_p$$ is sufficient for the set of sums of the rearranged series $\Sigma_\sigma f_n$ to be a closed linear set in $L_p$. 
@article{MZM_1973_14_1_a3,
     author = {E. M. Nikishin},
     title = {Rearrangements of series in $L_p$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {31--38},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_14_1_a3/}
}
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E. M. Nikishin. Rearrangements of series in $L_p$. Matematičeskie zametki, Tome 14 (1973) no. 1, pp. 31-38. http://geodesic.mathdoc.fr/item/MZM_1973_14_1_a3/