# Closed subspace of $L^1[0,1]$

The statement I need to prove is following. Let $S$ be a closed subspace of Lebesgue space $L^1[0,1].$ Assume that for every $f\in S$ there exists a number $p(f)>1$ such that $f\in L^{p(f)}[0,1].$ Then there exists a number $p>1$ such that $S\subset L^p[0,1].$

As far as I understand, this problem is related to Baire theorem. Hence, I write

$S=\bigcup^{\infty}_{n=1}( S\cap L^{1+1/n} [0,1])$

and conclude that for some $n$ closure of $S\cap L^{1+1/n} [0,1]$ has nonempty interior. But how can I proceed further to show that it lies in some $L^{1+1/m}$?

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Apply Baire category theorem to the closed sets $$F_n:=\{f\in S,\lVert f\rVert_{L^{1+n^{-1}}}\leqslant n\}$$ (these one are closed by Fatou's lemma).