Question: Given a probability space ($\Omega, \mathcal{F}, P$) and a measurable space ($S,\Sigma$), and an S-valued stochastic process { $X\_t : t \in T$ }(assume every $X\_t$ is i.i.d). What does a sample paths mean? From the definition, it's 'the function on $T$ to the range of the process which assigns to each $t$ the value $X\_t(\omega)$, where $\omega$ is a previously given fixed point in the domain of the process.

But if the $\omega$ is fixed, then the value of each $X\_t$ should be the same, which seems not true. What's the problem here?

Answer: 
It can help to *look at* some sample paths. From Bernt Øksendal's [Stochastic Differential Equations](https://rads.stackoverflow.com/amzn/click/com/3540047581):

[![Sample paths](https://i.stack.imgur.com/TYG9T.png)](https://i.stack.imgur.com/TYG9T.png)

The image shows five sample paths of a geometric Brownian motion process $\{X\_t\}\_{t\ge0}$. The paths are different functions of $t$ (which you may think of as representing *time*). Each one of them shows the values that $X$ takes over time under a specific outcome ($\omega\_1,\dots,\omega\_5$) in the sample space $\Omega.$

Notice by contrast that the expected value $E[X\_t]$ of the process is also drawn (as a smooth function of $t$) and that it is not a sample path; rather than being the result of any single outcome, it is obtained by "averaging" over all of them according to the law of the process.
Each random variable $X\_t$ is an $\mathcal F$-measurable function $\omega\mapsto X\_t(\omega)$ from $\Omega$ to $S$. The stochastic process $X$ may therefore be viewed as a function of two variables $\Omega\times T\ni (\omega,t)\mapsto X\_t(\omega)\in S$. For each fixed $\omega\in\Omega$ we have the partial function $t\mapsto X\_t(\omega)$; this is the *sample path* of $X$ corresponding to the sample point $\omega$.