# For stochastic processes, what is the difference between the sample space and the set of values that the random variables can take?

A stochastic process is defined as a collection of random variables defined on a common probability space (Ω,F,P), where Ω is a sample space, F is a σ-algebra, and P is a probability measure; and the random variables, indexed by some set T, all take values in the same mathematical space S, which must be measurable with respect to some σ-algebra Σ.

Now, unfortunately, this definition confuses me more than it enlightens me. I am confused by the difference between Ω and S, and F and Σ. It seems to me that the definition of a sample space is the set of all of possible outcomes/values, so the definitions of Ω and S identical to me.

Thanks for your effort! I have already searched this website to see if the question already exists, but I could not find it.

• Suppose $(\Omega,\mathscr{F})$ and $(T,\mathcal{T})$ are measurable spaces. A stochastic process with values in $T$ and indexed by $I$ is a function $X:\Omega\rightarrow T^I$ that is measurable. The measurable structure on $T^I$ is that of the product $\sigma$-algebra, that is the $\sigma$-algebra on $T^I$ generated by the sets of the form $A_{i_1}\times \ldots\times A_{i_n}$, where $A_{i_\ell}\in\mathcal{T}$. May 6, 2021 at 7:01

Let's consider tossing a coin. $$X_t$$ will denote the t-th toss. For the sake of simplicity I will only consider 2 coin tosses ($$t \le 2$$)

• We have elementary events: $$\omega$$ for example: $$(H, H), (H, T), ...$$

• Then the sample space is the set of all $$\omega$$ (\Omega). We also have events ($$A$$) which are asubset of $$\Omega$$.

• $$F$$ is a $$\sigma$$-algebra which contains measureable events. For example one possible choice of events is $$F=\{\emptyset, \Omega\}$$ where we can only measure these 2 trivial events.

• $$P$$ is the probability measure, which is a function $$\Omega \to [0,1]$$, it tells you the probability of each $$\omega$$

Up until now we did not talk about $$S$$ or $$\Sigma$$

• $$S$$ is some valued space, usually $$R \text{, or } R^n$$
• A random variable is measureable function $$\Omega \to S$$.
• Random variables generates a sigma algebra, which is the most simple sigma algebra for which it is measurable to. For example $$X_1$$ is measurable on the $$\sigma$$-algebra $$\sigma(X_1)=\big\{ \{(H,H), (H,T)\}; \{(T,H), (T, T)\}, \emptyset, \Omega \big\}$$

I hope this helps clarify things. I'd recommend to look into meassure theory and probability theroy before jumping to stochastic processes.