Definition of sample space

Question

From Rohatgi-Saleh's book on probability and statistics:

Def: The sample space of a statistical experiment is a pair $(\Omega,\mathcal S)$, where

(a) $\Omega$ is the set of all possible outcomes of the experiment.

(b) $\mathcal S$ is a $\sigma$-field of subsets of $\Omega$.

I wish to know why a $\sigma$-field $\mathcal S$ is associated with $\Omega$ in the definition of sample space of a random experiment. Is this just for convenience in defining any set $A\in\mathcal S$ as an event?

I know the definition of a $\sigma$-field and I'm aware why it is important in defining probability as a function. However, I don't understand why the concept of $\sigma$-field is required in defining the sample space. I thought we needed it because it is the domain of the event function $\mathbb{P}$ in Kolmogorov's axioms.

Question: Why is this definition more rigorous than simply saying that the sample space of a random experiment is the set of all possible elementary events (where I define elementary event as an outcome of a random experiment which cannot be decomposed into further outcomes)?

Answer 1

The basic intuition is that:

• $\Omega$ is the set of outcomes that can happen.
• $\mathcal S$, a $\sigma$-field of subsets of $\Omega$, represents what information is available. It represents what outcomes can be distinguished from each other. It is the set of events where an event is itself a set of outcomes.

You may not be able to tell certain outcomes apart, and these outcomes may be combined into a single event. This structure becomes especially useful when thinking about the arrival of new information over time. It's a useful mathematical structure for dealing with different information.

Example:

Let:

• $ww$ denote the outcome where the Cubs win the first two games of the World Series
• $wl$ denote the outcome where they win the first game but lose the second
• $lw$ denote the outcome where they lose the first game but win the second
• $ll$ denote the outcome where the Cubs lose both games

The sample space for the first two games is given by: $$\Omega = \{ww, wl, lw, ll\}$$

Before any games are played:

One possible $\sigma$-field is given by: $$ S_0 = \left\{ \left\{ \emptyset \right\}, \left\{ww, wl, lw, ll \right\} \right\} $$

This captures what is knowable before any game is played. You can't distinguish between any of the outcomes. And any random variable $X_0$ observable at time $t=0$ should map outcomes $ww, wl, lw, ll$ to the same value. This is captured formally with the notion of a measurable function.

After the first game:

After the first game is played, there is additional information, hence: $$ S_1 = \left\{ \left\{ \emptyset \right\}, \left\{ww, wl \right\}, \left\{lw, ll \right\}, \left\{ww, wl, lw, ll \right\} \right\} $$ That is, you can tell whether the Cubs won or lost the first game, but you can't tell apart $ww$ from $wl$! You can't tell who won the second game.

Let $C_1$ be a random variable denoting whether Cubs won game 1. $C_1$ is not measurable with respect to $S_0$ but it is measurable with respect to $S_1$.

Let $C_2$ be a random variable denoting whether the Cubs won game 2. $C_2$ is not measurable with respect to $S_1$. The preimage of $C_2(\omega) = \text{TRUE}$ is the set $\{lw, ww\}$, and that set isn't in $S_1$.

After the second game:

And finally you would have $S_2 = 2^{\Omega}$ (i.e. it's the powerset).

$$ S_2 = \left\{ \emptyset, \{ww\}, \{wl\}, \{lw\}, \{ll\}, \left\{ww, ll \right\}, \left\{ww, lw \right\}, \left\{ww, wl \right\}, \ldots \right\} $$

$\mathcal{S} = (S_0, S_1, S_2)$ is called a filtration.

Conclusion

$\Omega$ represents possible outcomes. $S$ is the set of events where each event is a set of outcomes. $S$ captures information in the sense of which outcomes are distinguishable. This distinction between what can happen vs. what is knowable, between outcomes and events, is extremely useful when information is revealed over time.