If the notes only discuss discrete valued RVs there is no problem. This looks like the case since from greping for "continuous" there are few results: https://github.com/ermongroup/cs228-notes/search?l=Markdown&...
Will be in trouble when considering a sequence of, say, independent random variables, say, as in coin flipping.
Will have trouble with P(A|B) and E[Y|X].
And the notation is still wrong.
But if do try to give the measure theory foundations, as the OP did, then at least don't make a mess out of it.
If don't want to get the measure theory right, then, sure, just leave out the foundations and start with events and random variables. The measure theory foundations are so powerful, so robust, so general that in practice it's tough to get into trouble. A place to get into trouble: In stochastic processes, take the union of uncountably infinitely many points, call that an event, and ask for its probability. Okay, then, don't do that.