In their "Probability review" at

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.