Characterization of Dirac Measure

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Let $x_0$ be a point in a set $X$ and $\delta_{x_0}$ the Dirac measure concentrated at $x_0$. Characterize the nonnegative real-valued functions on $X$ that are integrable over $X$ with respect to $\delta_{x_0}$ and the value of $\int_X f \delta_{x_0}$.