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Let $(X, d, \mu)$ be a metric measure space, with $\mu$ a Borel regularmeasure. In this paper, we prove that, if $u\inL^1_{{\mathop\mathrm{\,loc\,}}}(X)$ and $g$ is a Haj{\l}asz gradient of $u$,then there exists $\widetilde u$ such that $\widetilde u=u$ almost everywhereand $4g$ is a $p$-weak upper gradient of $\widetilde u$. This result avoids apriori assumption on the quasi-continuity of $u$ used in [Rev. Mat.Iberoamericana 16 (2000), 243-279]. As an application, an embedding of theMorrey-type function spaces based on Haj{\l}asz-gradients into thecorresponding function spaces based on upper gradients is obtained. We alsointroduce the notion of local Haj{\l}asz gradient, and investigate therelations between local Haj{\l}asz gradient and upper gradient.

Hajlasz Gradients Are Upper Gradients