Let $dx$ be the Lebesgue measure on $\mathbb R^d$. Let $u:\mathbb R^d\to{\mathbb R}\cup\{\infty\}$ is a non-negative and measurable function. The question is that, what are the conditions on $u$ so that $u$ is a density of a positive measure $\mu$, that is $d\mu(x)=u(x)dx$.If $u$ is finite on a positive measurable $E$, then is it true that $u$ is a density of a positive measure ?
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$\begingroup$As explained in the comments, every (measurable) function $u$ will do.
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