With the notation of the previous paragraph, observe that the measures we have defined are positive when the smooth metrized line bundles are semi-positive. Indeed, this means that the line bundles $\mathfrak{L}_{j}$ are numerically effective hence, as a consequence of the criterion Nakai–Moishezon, any subvariety of the special fiber has a nonnegative multidegree.

With basically the same argment that the one we sketched in the complex case, we conclude that the local height pairing extends by continuity when semi-positive metrized line bundles are approximated by smooth semi-positive metrized line bundles. By linearity, this extends the local height pairing to admissible metrized line bundles.