We define an inner product on a vector space of adelic measures over a number field. We find that the norm induced by this inner product governs weak convergence at each place of
K
K
. The canonical adelic measure associated to a rational map is in this vector space, and the square of the norm of the difference of two such adelic measures is the Arakelov–Zhang pairing from arithmetic dynamics. We find that the norm of a canonical adelic measure associated to a rational map is commensurate with a height on the space of rational functions with fixed degree. As a consequence, we show that the Arakelov–Zhang pairing of two rational maps
f
f
and
g
g
is, when holding
g
g
fixed, commensurate with the height of
f
f
.