No problem. For the
foundations I outlined,
can work just fine with
continuous functions,
measurable functions,
stochastic processes,
random variables taking
values on the real line,
in the complex plane,
in finite dimensional
real or complex vector
spaces with, say,
the usual topology,
Hilbert and Banach spaces,
etc. Can do multi-dimensional
Markov processes, and much more.

And you can have each point on the
real line an event. Fine.
But you just can't take
the uncountable union of
any set of such events and assume
that the result is also an event.

As for the event a random variable
takes a value >= 0? Fine.

Or, let the Borel subsets of the
real line be the smallest sigma algebra
that contains all the open sets,
e.g., all the open intervals.
Then for Borel set A and
real valued random variable X,
can ask for the probability
X is in A.

I believe you will find that
you will have a solid foundation
for what you want.

To see all this stuff, need more
than just the sparse definitions
and, instead, need an actual
text and maybe a course. Recently
looked at the on-line materials
from MIT and didn't see such a course.
Graduate probability is not
all that popular in the US;
stochastic processes in continuous
time is still less popular.

To study graduate probability,
I'd recommend a good
undergraduate major in pure math
with good coverage of, say,
W. Rudin, Principles of Mathematical
Analysis. Then good coverage of
linear algebra from more than
one of the best known texts.
Likely also spend as much time
as you can in Halmos, Finite
Dimensional Vector Spaces.
E.g., at one time, Halmos,
Rudin, and Spivak, Calculus
on Manifolds were the three
main texts for Harvard's
famous Math 55.

Get good with proving the theorems.

I also recommend Fleming,
Functions of Several Variables.

Then, sure, Royden, Real Analysis.
Couldn't be prettier.

If not in a hurry, then
the real half of Rudin's
Real and Complex Analysis.
Especially if you like Fourier
theory!

Then of the probability books,
I believe that the nicest, first
book is L. Breiman, Probability.
He wrote that before he went
consulting and came back and
did CART and random forests.

Next, K. Chung,
A Course in Probability Theory.
Next, J. Neveu, Mathematical
Foundations of the Calculus of Probability.
Then, Loeve, Probability Theory.

Loeve is huge -- mostly just
use it for reference or browse.
E.g., it has sufficient statistics
and stationary stochastic processes
(the EEs love that) IIRC
not in the other
books.

IIRC, both Breiman and Neveu
were Loeve students at Berkeley.

If do well with Breiman, then
for graduate probability,
likely can stop there.
Else, Chung will then be
fast and easy reading and
reinforce what you learned in
Breiman. Neveu is elegant;
my favorite, but deserve extra
credit for each workable exercise
you can find, not actually work,
you understand, just find! Sure,
some of the exercises are terrific,
half a course in a few lines of
an exercise. E.g., he has one
of those on statistical decision
theory or some such. And see the
Tulcea material in the back.

Then there's more that you can
do on stochastic processes,
potential theory via Brownian
motion,
e.g., for mathematical finance,
stochastic optimal control,
and more.