  We present the theory of weak gravitational lensing in cosmologies with
generalized gravity, described in the Lagrangian by a generic function
depending on the Ricci scalar and a non-minimally coupled scalar field. We work
out the generalized Poisson equations relating the dynamics of the fluctuating
components to the two gauge invariant scalar gravitational potentials, fixing
the new contributions from the modified background expansion and fluctuations.
We show how the lensing observables are affected by the cosmic expansion as
well as by the presence of the anisotropic stress, which is non-null at the
linear level both in scalar-tensor gravity and in theories where the
gravitational Lagrangian term features a non-minimal dependence on the Ricci
scalar. We derive the generalized expressions for the convergence power
spectrum, and illustrate phenomenologically the new effects in Extended
Quintessence scenarios, where the scalar field coupled to gravity plays the
role of the dark energy.
