  Gravitational wave memory is a nonoscillatory correction to the gravitational
wave strain predicted by general relativity, which has yet to be detected.
Within general relativity, its dominant component, known as the null memory,
can be understood as arising from the backreaction of the energy carried by
gravitational waves, and therefore it corresponds to a direct manifestation of
the nonlinearity of the theory. In this paper, we investigate the null-memory
prediction in a broad class of modified gravity theories, with the aim of
exploring potential lessons to be learned from future measurements of the
memory effect. Based on Isaacson's approach to the leading-order field
equations, we in particular compute the null memory for the most general
scalar-vector-tensor theory with second-order equations of motion and vanishing
field potentials. We find that the functional form of the null memory is only
modified through the potential presence of additional radiative null energy
sources in the theory. We subsequently generalize this result by proving a
theorem that states that the simple structure of the tensor null-memory
equation remains unaltered in any metric theory whose massless gravitational
fields satisfy decoupled wave equations to first order in perturbation theory,
which encompasses a large class of viable extensions to general relativity.
