weakly $p-$summable sequences in $L_{\infty}(\Omega, \mu)$

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Let $1\leq p<\infty$ and $(\Omega, \mu)$ be a measure space and $f_{1},\ldots,f_{m}\in L_{\infty}(\Omega, \mu)$. What is the definition of weakly $p-$summable sequences $(f_{j})_{j=1}^m$ in $L_{\infty}(\Omega, \mu)$ ?