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Existence of a Borel measurable function below any positive function

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Mario
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Existence of a Borel measurable function

Let $f\colon \mathbb R\to (0,\infty)$ be a function taking positive values. Does there exist a Borel measurable function $g\colon \mathbb R\to (0,\infty)$ taking positive values as well such that $g(x)\leq f(x)$ for all $x\in\mathbb R$?