What is an example of a sample space in machine learning?

Let $$X$$ denote a random variable. Then from a rigorous mathematical perspective (books such as Durrett, Feller, Kolmogorov, etc.),

$$X$$ is a function. $$X: \Omega \to \mathbb{R}^n$$.

Domain of the function is the sample space, $$\Omega$$

Range is a value in $$\mathbb{R}^n$$.

In supervised learning, let $$X, Y$$ denote the random variable corresponding to the data $$x_n \in \mathbb{R}^n$$ and target $$y_n \in \mathbb{R}$$ respectively.

Then $$X$$ maps from a sample space into a piece of data $$x_n$$. And $$Y$$ maps from a sample space into a piece of target, $$y_n$$.

With this, the notation, such as $$p_{X|Y}(x_n|y_n) = \Pr[X = x_n| Y = y_n]$$

is completely well defined.

Then we can start talking about things like logistic regression, i.e.,

$$p_{X|Y}(x_n|y_n) = \Pr[X = x_n| Y = y_n] = \text{logit}(y_nw^Tx_n) \in (0,1)$$

etc.

But what is the sample space $$\Omega$$? Any example would be much appreciated!