L^p norms

Prove or disprove that to every positive function g on (0, infty) such that
g(p)-->infty as p-->infty, there exists a Lebesgue measurable function f on
(0,1) such that ||f||_p --> infty as p-->infty and ||f||_p <= g(p) for
sufficiently large p.
Note: Such an f , if exists, cannot be essentially bounded.


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