why does one define the Ito integral as the $L^2$ limit, although it can be shown by Doob's martingale inequality and Borel-Cantelli lemma that there exists a t continuous version, which is constructed as almost sure limit. So why not define the Ito integral as the continuous almost sure limit?
Thank you in advance
The key advantage of Ito integrals is that they are martingales. For this, a.s. convergence would not be enough in general. You don't quite need $\mathcal{L^2}$ convergence, but something almost as strong implying some boundedness. For a reference and good discussion, maybe compare Øksendal; also this question at MathOverflow.