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.
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