How do people in probability deal with abstract or uncountably infinite sample spaces?

Question

In elementary probability class, we are taught simple examples such as,

Tossing a coin, generating the sample space $S = \{H,T\}$.

Rolling a die, generating the sample space $S = \{1, 2, 3, 4, 5, 6\}$

Simple examples like this. Here, it is easy to define random variable $X$ that maps each point $w \in S$ into a real number $X(w) \in \mathbb{R}$.

However, in real life, we might deal with the following examples.

Generate a picture using a neural network, the sample space is the space of all pictures $S = \{\text{every picture}\}$

or

mapping the thoughts of a human into an EEG electrical signal, $S = \{\text{thoughts}\}$.

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In these cases, is it still possible to use probability at all?

The sets are "abstract" or in more concrete terms uncountably infinite.

If we cannot even describe what this set is, how can we proceed to describe a random variable $X$? How can we even say that one exists?

We have no idea how this random variable even works or how it should be defined? Then how do we describe distributions of random variables?

Answer 1

In fact, a majority of modern research in probability theory is concerned with uncountable sample spaces. Look no further than the two most fundamental objects in all of probability: the uniform $[0,1]$ random variable, and the Brownian motion $B_t$. The former has sample space $\Omega=[0,1]$ and the latter has sample space $C_0([0,1])$, the space of all continuous functions on $[0,1]$ started from $0$.

Mathematicians are accustomed to working with objects like $\mathbb R$ and function spaces, and at this point have a very precise understanding of these objects. In many regards (especially concerning calculations) it is no more difficult than working with a finite sample space - and sometimes easier, since the infinite objects often follow simpler patterns.

And of course, it all uses the apparatus of measure theory, as already mentioned in the comments.

Being able to literally map an abstract concept into physical life has little bearing on the mathematical reasoning required... and in a sense that is the point of mathematics.

Furthermore, I disagree with your assessment that we cannot describe a set which is uncountably infinite. It is no different than saying that you cannot describe a glass of water, since you cannot manage to count each molecule of water in your lifetime...

Answer 2

You are conflating two ways in which your simple "roll the die" examples can be extended.

One extension of discrete random variables is to construct continuous random variables. A simple example is a variable $X$ that is uniformly distributed on the interval of numbers $[0,1].$ There are uncountably many possible values of $X,$ but we manage to construct a distribution nevertheless.

Another example of a continuous random variable is a variable $Y$ that has a normal distribution with mean $\mu$ and variance $\sigma^2$. Such variables are very frequently used in practical applications.

The other way to extend random variables is to have not just a single variable with a single random numerical value, but to have an vector or other structure containing multiple random variables. You would want something like that for either the example of a cartoon image or the example of an EEG signal.

By the way, an EEG signal is not the same thing as a thought. Nobody as far as I know has a mathematical model able to distinguish any thought among all possible thoughts, never mind a way to make a probability distribution over that model. What the researchers in the EEG experiment did was to hook up a machine to a person that extracts just a few electrical signals from the person's body (which have something to do with their thoughts, but are not the entire thoughts), and then ignore most of the information that might be found in those signals and look at just a few properties (which can be described by a vector of numbers).