I know that $\int_0^T g(W_t,t)dW_t$ is a continuous martingale, but I can't get what it represent exactly and can't find any justification anywhere. It's written every where that it's fundamental in modern stochastic calculus but nowhere explain why is it so important. I can't really connect it to the usual integral I know.
I read somewhere that $(W_t)_t$ is a measure on continuous function... So maybe, if $A\subset \mathcal C^0$, then $\mu(A)=\int_A dW_t$ is the measure of $A$ ? And so ?