Suppose that $W_t$ is a Brownian motion. Consider
$$X_t = X_0 + \int_{0}^t f(s, \omega)ds + \int_{0}^t g(s, \omega)dW_s$$
The question is about the second term $\int_{0}^t f(s, \omega)ds $.
We usually imply that the integral $\int_{0}^t f(s, \omega)ds $ is pathwise (it means that for any fixed $\omega = \omega_0$ we have $\int_{0}^t f(s, \omega_0)ds$). But what exacltly does this term mean?
In books "Topics in stochastic process" by R.B. Ash, M. F. Gardner, (p.226) and "Theory and Statistical Applications of Stochastic Processes" (p.203) by Y. Mishura and G. Shevchenko it's said that $P( \int_{0}^t |f(s, \omega)|ds < \infty) = 1$ and $f(s, \omega)$ progressively measurbale.
In the book "Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory (Problem Books in Mathematics)" (p.194) it's said that $f(s, \omega)$ is measurable, adapted to filtration and $P( \int_{0}^t |f(s, \omega)|ds < \infty) = 1$.
In the book "Introduction to Stochastic Intergation" by Hui-Hsiung Kuo, 2006, p.95, (in a special case $\int_{0}^t f''(W(s))ds$) it's said that the $ \int_{0}^t f(s, \omega)ds$ is a Riemann integral for each sample path of $W_s$.
The question appeared in connection with the next fact. In the book: "Theory of Random Processes in Examples and Problems." by B. M. Miller and A. S. Pankov it's said that the integral $\int_{0}^t f(s, \omega)ds $ is $L_2$ intergral, i.e. the $L_2$-limit of $\sum_{i=1}^n f(s_i, \omega) (t_i - t_{i-1})$.
So, we may create several defintions of $\int_{0}^t f(s, \omega)ds$.
Defintion 1: It's Lebesgue pathwise integral + we imply that $P( \int_{0}^t |f(s, \omega)|ds < \infty) = 1$ and $f(s, \omega)$ progressively measurbale.
Defintion 2: It's Lebesgue pathwise integral + we imply that $P( \int_{0}^t |f(s, \omega)|ds < \infty) = 1$ and $f(s, \omega)$ is measurable and adopted to filtration.
Defintion 3: It's $L_2$-limit of $\sum_{i=1}^n X_{s_i} (t_i - t_{i-1})$ (Riemann $L_2$ integral).
Defintion 4a: It's pathwise Riemann intergal for all $\omega$.
Defintion 4b: It's pathwise Riemann intergal for $\omega \in \Omega^*: P(\Omega^*) = 1$.
I got used to defintion 1. The question is why another definitions do not work? (or if somebody disagree with defintion 1 then the question is "why? + which definition works? + why does is work?")
Addition: Is there any natural reason to avoid, for example, the usage of definition $i$, $i \ge 2$? Is there any important theorem, that works for definition $1$, but doens't work with definition $i$? I want to understand, why we should prefer one definition to another. If some definition is "bad" in some sense, then why?