n >= 2 / p
cubic inches. Then the probability that the particle is in those n cubic inches is
np >= 2 > 1
greater than 1, a contradiction. Done.
One well considered and informed explanation is that the physics community abuses its students.
"A synthetic approach to Markov kernels..."
A probability theory that accommodates the concept of "picking a random even number from all integers" can be valuable, isn't compatible with measure theory but is easy to grasp intuitively. When the mathematical tools aren't good enough you still want to be able to reason about concepts, which is why the tools are developed to begin with. Fortunately almost all things are conceptually tractable when dealing with finite space or quantities; if a statement can be transformed to something rigorous (if numerically imprecise) by forcing boundedness it's not too terrible to speak in unbounded terms when the physical world is what you want to model. I can understand why physicists don't want to be burdened with too much about rigor - they can afford the small risk they're wrong sometimes, but can't afford to slow down their search for new discoveries, when so many questions remain unanswered.
The MIT quantum mechanics lecture I mentioned keeps using random -- struggles with it, says it over and over as if that would make it more clear. In that lecture, the meaning of random is not clear.
What that prof means is likely
"the values of some random variables that are independent and identically distributed (i.i.d.)."
Then we can have some random variables that take values only on the positive, even integers. But, sure, they can't be uniformly distributed. So, with the values of such random variables, we can get some even integers selected independently.
In good treatments of probability, e.g., (from my bibliography file, with TeX markup)
Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco.
independence is very nicely developed, e.g., to handle even a set of uncountably infinitely many independent random variables.
As I recall, Neveu was a Loeve student.
That book my Neveu is my candidate for the most elegantly written math book of all time.
But the development of independence is not trivial.
I would expect that independence is needed at times in physics, especially in quantum mechanics, but that only a small fraction of physics profs have studied independence much like what is in Neveu. One result is some confusion between independent and uncorrelated. When I'm watching a physics lecture and see such confusion, I stop taking the physics prof seriously and turn to something more serious, say, an old Marilyn Monroe movie!
But sure, if you're assuming an unbounded space with finite measure, a uniform density across that space must be identically zero everywhere.
Not a probability density.
