  We deal with the solution of a generic linear inverse problem in the Hilbert
space setting. The exact right hand side is unknown and only accessible through
discretised measurements corrupted by white noise with unknown arbitrary
distribution. The measuring process can be repeated, which allows to reduce and
estimate the measurement error through averaging. We show convergence against
the true solution of the infinite-dimensional problem for a priori and a
posteriori regularisation schemes as the number of measurements and the
dimension of the discretisation tend to infinity under natural and easily
verifiable conditions for the discretisation.
