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Given a numerical function $\\varphi$ on $X$ which is locally lower bounded, let \\begin{equation*}\n  J_\\varphi(x):=\\sup\\{\\int^\\ast \\varphi\\,d\\mu(x)\\colon \\mu\\in \\mathcal J_x(X)\\}, \\qquad x\\in X, \\end{equation*} where $\\mathcal J_x(X)$ denotes the set of all Jensen measures $\\mu$ for $x$, that is, $\\mu$ is a compactly supported measure on $X$ satisfying $\\int u\\,d\\mu\\le u(x)$ for every hyperharmonic function on $X$. 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