In this paper, we extend the well-known Frostman lemma by showing that for any subset E\documentclass[12pt]{minimal}
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\begin{document}$$E$$\end{document} of [0,1]\documentclass[12pt]{minimal}
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\begin{document}$$[0, 1]$$\end{document} and α>0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha >0$$\end{document}, if the α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}-Hausdorff measure of E\documentclass[12pt]{minimal}
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\begin{document}$$E$$\end{document} is positive, then there exist a non-zero Borel measure μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} on [0,1]\documentclass[12pt]{minimal}
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\begin{document}$$[0, 1]$$\end{document}, a constant C>0\documentclass[12pt]{minimal}
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\begin{document}$$C>0$$\end{document} and a subset E0\documentclass[12pt]{minimal}
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\begin{document}$$E_0$$\end{document} of E\documentclass[12pt]{minimal}
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\begin{document}$$E$$\end{document} such that μ(I)≤C|I|α\documentclass[12pt]{minimal}
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\begin{document}$$\mu (I) \le C \vert I \vert ^{\alpha }$$\end{document} for any interval I\documentclass[12pt]{minimal}
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\begin{document}$$I$$\end{document} and E0\documentclass[12pt]{minimal}
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\begin{document}$$E_0$$\end{document} is dense in the support of μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}. Under an additional condition on E0\documentclass[12pt]{minimal}
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\begin{document}$$E_0$$\end{document}, we show that μ(B)=μ[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$\mu (B) = \mu [0, 1]$$\end{document} for any Borel subset B\documentclass[12pt]{minimal}
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\begin{document}$$B$$\end{document} containing E\documentclass[12pt]{minimal}
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\begin{document}$$E$$\end{document}. Using the notion of Choquet integral, we extend the notion of capacitarian dimension to arbitrary subset of [0,1]\documentclass[12pt]{minimal}
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\begin{document}$$[0, 1]$$\end{document} and prove a generalisation of Frostman’s theorem.
