Let $(X,M,\mu)$ be a measure space with $\mu(X)=1$ and $f_n\in\mathcal{L}^2(X)$ such that there exists an $a>0: 1\leq\left\lVert{}f_n\right\rVert_2\leq{}\frac{1}{a}\left\lVert{}f_n\right\rVert_1$ for all $n\geq{}1$. Show that $\mu(\{x\in{}X:|f_n(x)|\geq{}a/2 \text{ for infinite values of } n\})\geq{}a^2/4$.

Observations: The condition becomes $1\leq{}\int_X(f_n)^2d\mu\leq{}(\frac{1}{a}\int_X|f_n|d\mu)^2$ and setting $E_n=\{x\in{}X:|f_n(x)|\geq{}a/2\}$ we observe that we need to show that $E=\{x\in{}X:|f_n(x)|\geq{}a/2 \text{ for infinite values of } n\} =\displaystyle\limsup_{n\to\infty}E_n$

Obviously $a\leq{}1$ since $\int_X(f_n)^2d\mu\geq{}(\int_X|f_n|d\mu)^2$ by Jensen's inequality.

Any ideas about how to proceed?


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