http://ermongroup.github.io/cs228-notes/preliminaries/probab...
I see two problems:
(1) First Problem -- Sample Space
Their definition of a sample space is
"The set of all the outcomes of a random experiment. Here, each outcome ω can be thought of as a complete description of the state of the real world at the end of the experiment."
The "complete description" part is not needed and even if included has meaning that is not clear.
Instead, each possible experiment is one trial and one element in the set of all trials Ω. That's it: Ω is just a set of trials, and each trial is just an element of that set. There is nothing there about the outcomes of the trials.
Next the text has
"The sample space is Ω = {1, 2, 3, 4, 5, 6}."
That won't work: Too soon will find that need an uncountably infinite sample space. Indeed an early exercise is that the set of all events cannot be countably infinite.
Indeed, a big question was, can there be a sample space big enough to discuss random variables as desired? The answer is yes and is given in the famous Kolomogorov extension theorem.
(2) Second Problem -- Notation
An event A is an element of the set of all events F and a subset of the sample space Ω.
Then a probability measure P or just a probability is a function P: F --> [0,1] that is, the closed interval [0,1].
So, we can write the probability of event A by P(A). Fine.
Or, given events A and B, we can consider the event C = A U B and, thus, write P(C) = P(A U B). Fine.
But the notes have P(1,2,3,4), and that is undefined in the notes and, really, in the rest of probability. Why? Because
1,2,3,4,
is not an event.
For the set of real numbers R, a real random variable X: Ω --> R (that is measurable with respect to the sigma algebra F and a specified sigma algebra in R, usually the Borel sets, the smallest sigma algebra containing the open sets, or the Lebesgue measurable sets).
Then an event would be X in {1,2,3,4} subset of R or the set of all ω in Ω so that X(ω) in {1,2,3,4} or
{ω| X(ω) in {1,2,3,4} }
or the inverse image of {1,2,3,4} under X -- could write this all more clearly if had all of D. Knuth's TeX.
in which case we could write
P(X in {1,2,3,4})
When the elementary notation is bad, a bit tough to take the more advanced parts seriously.
A polished, elegant treatment of these basics is early in
Jacques Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, 1965.
Neveu was a student of M. Loeve at Berkeley, and can also see Loeve, Probability Theory, I and II, Springer-Verlag. A fellow student of Neveu at Berkeley under Loeve was L. Breiman, so can also see Breiman, Probability, SIAM.
These notes are from Stanford. But there have long been people at Stanford, e.g., K. Chung, who have these basics in very clear, solid, and polished terms, e.g.,
Kai Lai Chung, A Course in Probability Theory, Second Edition, ISBN 0-12-174650-X, Academic Press, New York, 1974.
K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, Second Edition, ISBN 0-8176-3386-3, Birkhaüser, Boston, 1990.
Kai Lai Chung, Lectures from Markov Processes to Brownian Motion, ISBN 0-387-90618-5, Springer-Verlag, New York, 1982.