TITLE: Ito integral almost sure and $L^2$ limit
QUESTION [8 upvotes]: 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

REPLY [3 votes]: 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.