Perhaps one of the most clever and important abstractions in probability theory is that of having a measure space $(\Omega, \Sigma, \rmP)$ that we never explicitly study. Instead, we implicitly study the object by analyzing the measurable functions $X: \Omega \rightarrow \bbX$ into other measurable spaces $(\bbX, \calX)$ with known structure (like being Borel, finite, etc.). At first, this practice seems rather odd—why do we never care to explain exactly what $(\Omega, \Sigma, \rmP)$ is? This question is quickly answered when we think of the objective of applying probability, which for the purposes of this post will be as follows.
We use probability to model and analyze the unpredictable values of all sorts of quantities $X$ we would measure in the real world.
For instance, if we have to roll $5$ dice, the unpredictable value of each die could be encoded with functions $X_1, \ldots, X_5: \Omega \rightarrow \{1, \ldots, 6\}$ defined on a probability space $(\Omega, \Sigma, \rmP)$. With such a setup, we can concisely denote probabilities of events, like the first dice rolling no smaller than a 5 $$ \rmP(X_1 \geq 5) = \rmP\big( X_1^{-1}\{5, 6\} \big), $$ we are able to quickly define notions like independence, $$ \rmP\big( X_1 \in A, X_2 \in B \big) = \rmP\big( X_1 \in A \big) \rmP\big( X_2 \in B \big) $$ and we may also use algebra and composition to encode related unpredictable outcomes, like the number of $2$'s rolled. $$ \sum_{i=1}^5 1_{\{2\}}(X_i): \Omega \rightarrow \{0, \ldots, 5\} $$ Any inference we could make with the outcomes of $5$-dice rolls $x_1, \ldots, x_5 \in \{1, \ldots, 6\}$ would be an expression $f(x_1, \ldots, x_5)$ like above, and so we may immediately model it as a random object $f(X_1, \ldots, X_5)$. This is massively convenient when compared to studying $(\Omega, \Sigma, \rmP)$ directly.
Let's actually verify this by doing the same model entirely from the perspective of studying $(\Omega, \Sigma, \rmP)$. The smallest space $(\tilde\Omega, \tilde\Sigma)$ which measures these $5$ dice rolls is the following product space.
$$\tilde\Omega = \{ 1, \ldots, 6 \}^5, \quad \tilde\Sigma = 2^{\tilde\Omega}$$By smallest, I mean to suggest that any $(\Omega, \Sigma, \rmP)$ that equips the $5$ dice rolls will immediately induce a measure $\tilde\rmP$ on $(\tilde\Omega, \tilde\Sigma)$ as follows,
$$\tilde\rmP(A_1 \times \cdots \times A_5) = \rmP\big( X_1 \in A_1, \ldots, X_5 \in A_5 \big)$$and so one may always push forward the model $(\Omega, \Sigma, \rmP)$ to $(\tilde\Omega, \tilde\Sigma, \tilde\rmP)$ via $(X_1, \ldots, X_5)$. With a specific measure space $(\tilde\Omega, \tilde\Sigma, \tilde\rmP)$ in mind, let us now consider how we would model the notions we had above. The first of which, denoting the probability of a specific event, looks far less concise now.
$$ \rmP\big(\{5, 6\} \times \{1, 2, 3, 4, 5, 6\} \times \cdots \times \{1, 2, 3, 4, 5, 6\}\big)$$It takes us a moment to realize we are talking about the first die rolling no smaller than a $5$; we need to fill $\{1, 2, 3, 4, 5, 6\}$ for the dice we otherwise don't care about. Independence of the first two die rolls now amounts to $$ \tilde\rmP\big(A \times B \times \{1, 2, 3, 4, 5, 6\} \times \cdots \times \{1, 2, 3, 4, 5, 6\}\big) = \tilde\rmP_1(A) \tilde\rmP_2(B) $$ for measures $\tilde\rmP_1, \tilde\rmP_2$ on $(\{1, 2, 3, 4, 5, 6\}, 2^{\{1, 2, 3, 4, 5, 6\}})$. Meanwhile, the events associated to the number of $2$'s rolled are now captured by the following measure. $$ A \mapsto \int_{\tilde\Omega} 1_A \Big( \sum_{i=1}^5 1_{\{2\}}(\omega_i) \Big) {\rm d}\tilde\rmP(\omega_1, \ldots, \omega_5) $$ The same can be said about an abstract inference $f$. $$ A \mapsto \int_{\tilde\Omega} 1_A \big( f(\omega_1, \ldots, \omega_5) \big) {\rm d}\tilde\rmP(\omega_1, \ldots, \omega_5) $$
The reason we keep $(\Omega, \Sigma, \rmP)$ abstract is merely simplification. In probability, we study objects $X: \Omega \rightarrow \bbX$ we'd observe as quantities $x \in \bbX$ in some measurable space $(\bbX, \calX)$. Our inferences $f(x) \in Y$ are now immediately modeled as subsequent random objects $f(X) = f \circ Y: \Omega \rightarrow \bbY$. As such, any deterministic thing we would do with an observation can itself be understood by our probability model. This allows all of our actions—before we even commit them—to fall victim to our meticulous probabilistic investigation 🤓.