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This is nothing but [[Markov's inequality]]. The converse is not true. For example, the function 1/''x'' belongs to ''L''<sup>1,''w''</sup> but not to ''L''<sup>1</sup>.
Similarly, one may define the [[Lp space#Weak Lp|'''weak <math>L^p</math> space''']] as the space of all functions ''f'' such that <math>|f|^p</math> belong to ''L''<sup>1,''w''</sup>, and the '''weak <math>L^p</math> norm''' using
:<math>\|f\|_{p,w}=\|\,|f|^p \|_{1,w}^{1/p}.</math>
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