No, he's just being sloppy.
That can be a real pain for
students trying to learn.
But, be warned: A lot of
people work with probability,
but only a tiny fraction ever
had a course in graduate
probability. So, eventually
have to learn to put up with,
and sometimes rewrite, some of
what is written that is not
very precise.
Here are the accepted definitions:
Take a non-empty set, usually
denoted by Omega, and call it the
sample space or the set of trials.
Take a collection of subsets of
Omega, usually denoted by script F,
and call it the set of events.
Have at least enough subsets of
Omega in the set of events script F
so that script F will be a sigma algebra.
So, script F has to have as an element
the empty set,
be closed under relative complements,
and be closed under countable unions.
We want all that for probability
theory. We want countably
infinite so that we can discuss,
say, the event that a coin never
or always
comes up heads. We don't want
uncountably infinite because
it would create a big mess
in the theory.
Then the ordered pair (Omega, script F)
is called a measurable space -- it
doesn't have a measure yet but
soon will.
On that measurable space, define
a positive measure P so that
P( Omega ) = 1.
Then P is a
probability measure on the
measurable space.
A probability space is
the triple ( Omega, script F, P ).
Intuitively,
a measure assigns to each
event A in script F a
non-negative real number P(A).
A signed measure permits
negative values. And
of course at times want
a measure that yields complex
numbers.
Measure theory is in, say,
P. Halmos, Measure Theory.
The back of Halmos has a
really nice introduction to
probability theory, the
Kolmogorov three series
theorem in stochastic processes,
etc. For more in stochastic
processes there is, of course,
the classic J. Doob, Stochastic
Processes. Halmos was a Doob
student and then served as
an assistant to von Neumann
at the Institute for Advanced
Study. From that he wrote
Halmos, Finite Dimensional
Vector Spaces, a finite
dimensional introduction to
Hilbert space. Later at
University of Chicago
Halmos worked in mathematical
statistics and made the
fundamental contribution to
sufficient statistics.
IIRC, it was at Chicago that
Halmos wrote Measure Theory.
Great details on measure theory
and much more
are in, say, H. Royden, Real
Analysis and W. Rudin, Real
and Complex Analysis.
To
narrow measure theory
to probability theory,
use the famous texts by the
authors I listed, Loeve, ....
Royden and Rudin are
terrific prerequisites to
Loeve, etc.
Commonly the sigma algebra of
events script F is the smallest
sigma algebra that contains
as a subset a topology, that is,
a collection of open sets,
on the
sample space Omega.
That's all standard, beginning
stuff in graduate probability --
ah, the only kind should bother
with anyway!