Quantum measurements and the associated state changes are properly described in the language of instruments. We investigate the properties of a time continuous family of instruments associated with the recently introduced class of general Gaussian non-Markovian stochastic Schrödinger equations. In this article we find that when the covariance matrix for the Gaussian noise satisfies a particular δ-function constraint, the measurement interpretation is possible for a class of models with self-adjoint coupling operator. This class contains, for example the spin-boson and quantum Brownian motion models with colored bath correlation functions. Remarkably, due to quantum memory effects the reduced state, in general, does not obey a closed form master equation while the unraveling has a time continuous measurement interpretation.