Q: How should I understand the probability space $(\Omega, \mathcal{F}, P)$? What does "hidden" mean? I thought I understood the measure theoretic concept of a probability space, but yesterday I realized that I really don't.  Here's what I mean:


*

*I thought we could think of $\Omega$ as the set of outcomes of a certain experiment.  So before we can even define the triple $(\Omega, \mathcal{F}, P)$, we would need to think about an experiment.  Now, each $\omega \in \Omega$ was supposed to be a possible outcome of the experiment, and the randomness thus comes from the fact that different outcomes can occur with different frequencies over time.  These long term frequencies define $P$ for us.

*So, if we consider the experiment of flipping a coin once, then $\Omega = \{ H, T\}$, since we only have those possible outcomes.

*Ok, but in the discussion of Brownian motion (or the diffusion of a particle across some environment like dye in water), we are told to think of each $\omega \in \Omega$ as a particle.  For example, each $\omega$ is a particle of pollen.  And the stochastic process $X_{t}(\omega)$ gives us, for each fixed $\omega$, the path that that particle $\omega$ follows.  But then what is the experiment we started with in order to get the set $\Omega$?  How are the particles that each $\omega$ represents the outcomes of some experiment?  And where does the randomness come from?  It seems like the randomness comes from the path the particle takes after we've already chosen it, so how can each particle be considered the outcome of some experiment?  Which experiment?

Finally, after reading the above (and in addition to the above questions), I'd really like to understand why in many times, the probability space $(\Omega, \mathcal{F}, P)$ is often "hidden" or not explicit.  Can anyone explain this intuitively or give me some useful examples where the probability space is hidden?  Shouldn't it always be explicitly defined (since math is always explicit)?


A: You should think of $\Omega$ more as a set of all possible futures.  So for the coin flipping, there are some futures where the coin comes up heads and some where it comes up tails.  You could let $\Omega$ be $\{H,T\}$, but you could also let it be the half-open interval $(0,1]$ with Lebesgue measure and say that $(0,1/2]$ is the set where the coin turns up heads and $(1/2,1]$ is the set where the coin turns up tails.  
The reason for preferring $(0,1]$ to ${H,T}$ is that it allows us to add in new random variables easily.  If we wanted to add a separate, independent coin toss to the first model, we'd be stuck.  But with $(0,1]$ we can do it (even though it's a bit messy):
$$\begin{array}{c|c|c|} 
 & \text{First coin heads} & \text{First coin tails} \\ \hline
\text{Second coin heads} & (0,1/4] & (1/4,1/2] \\ \hline
\text{Second coin tails} & (1/2,3/4] & (3/4,1] \\ \hline
\end{array}$$
The choice of the probability space itself doesn't really matter.  Normally, we set it to be $(0,1]$ (since a theorem tells us that $(0,1]$ is big enough to be the underlying measure space of most things we want to model, including countably infinite sequences of continuous random variables) and then forget about it.  It just happens that the language of measures captures our intuition for probability really well, and in order to use measures we need a measure space.  Unfortunately, there's no intuitive interpretation of that measure space, which is why it's often glossed over when we're talking about probability.
