Hello I am new to Ito integrals. I know how the Ito integral is defined and that it is not defined by pathwise convergence but in $L^2$ instead. But I think I actually did not quite get the point. How do I understand an expression like the following identity?
$$\int_0^t Bs \text{d}Bs = \frac{1}{2}B_t^2 - \frac{1}{2}t$$
Does it mean that for every $\omega$,
$$\Big(\int_0^t Bs \text{d}Bs\Big)(\omega) = \Big(\frac{1}{2}B_t^2 - \frac{1}{2}t \Big)(\omega) \text{ }?$$
In all my books the stochastic integrals are written without any $\omega$'s that's why I am asking where they would be...