 Solution

For simple predictable integrands, the Itô integral is a continuous square-integrable martingale and the Itô isometry givesApproximate a general square-integrable predictable integrand by simple ones. The resulting integral martingales are Cauchy in terminal $L^2$ norm. Completeness of continuous square-integrable martingales guarantees a unique continuous martingale limit, independent of the approximation, which defines the integral on the completed integrand space.