# Is stochastic integral the same as Lebesgue integral of Banach-valued function?

I'm reading a book about stochastic processes and I've come to "stochastic integration." In this section stochastic processes are basically functions $$X:[0,\infty)\rightarrow L^2(\Omega, \mathbb{C}).$$ I already know how to integrate functions from a measure space to a Banach space. Is this basically what this stochastic integral is doing?

• A stochastic process is a collection of random variables indexed by a totally ordered set, e.g. $[0,\infty)$. In other words, a stochastic process $X=\{X_t:t\in[0,\infty)\}$ consists of (uncountably) infinitely many random variables $X_t$. A stochastic integral of the form $\int f(s)\ \mathsf dX_s$ is not the same as a Bochner integral, as you are integrating with respect to a stochastic process, not a measure. – Math1000 May 18 at 22:53