The definition of the convergence of a sequence of measures is convergence of all integrals against a given continuous compactly supported function. In applications, however, it can be desirable to integrate against more general functions. The inductive formula (1.2.1) for the local height pairing in the complex case, is such an example, as is the interpretation of Mahler measures of polynomials as (the archimedean component of) heights. However, its analogue (Equation 1.3.1) a priori holds only when $\log\left\|{s_{0}}\right\|^{-1}$ is continuous, that is when the section $s_{0}$ has no zeroes nor poles.

The fact that it still holds in the archimedean case is a theorem of Maillot [43] building on the theory of Bedford–Taylor. We proved in [19, Th. 4.1] that this relation holds in the ultrametric case too. The proof (valid both in the ultrametric and archimedean cases) works by induction, and ultimately relies on an approximation lemma according to which any semi-positive Green function $g$ for a divisor $D$ is an increasing limit of smooth functions $(g_{n})$ such that, for any $n$, $g-g_{n}$ is a semi-positive Green function for $D$. In fact, it suffices to pose $g_{n}=\min(g,n\log\left|{\pi}\right|^{-1})$ ; then, $g-g_{n}=\max(0,g-n\log\left|{\pi}\right|^{-1})$ is the maximum of two semi-positive Green functions, hence is semi-positive. (In the archimedean case, one needs to further regularize $g_{n}$ ; see [19] for details.)

The symmetry of the local height pairing then implies the following analogue of the Poincaré–Lelong formula. When $\overline{L}$ is the trivial line bundle, with the metric defined by an admissible function $\varphi$, the factor $c_{1}(\overline{L})$ will be written $\mathop{\mathrm{d}\mathrm{d}^{c}}\varphi$, by analogy to the complex case.