I'm learning Ito calculus and integrals with respect to BM are defined but what about integrals of stochastic processes (like BM itself) with respect to Lebesgue measure?
It is true that $\int\limits_0^tW(s)ds=\lim\limits_{\Delta\rightarrow0}\sum\limits_{j=1}^{m_\Delta}W(t_{j-1})(t_j-t_{j-1})$ where $0=j_0<j_1<\dots<j_m=t$ is a partition of $[0,t]$ with $\Delta=\min_j|t_j-t_{j-1}|$ but is it expressible as a stochastic process $(X(t))_{t\in[0,T]}$ in a cleaner way than above?