Is there a uniform distribution over the real line?

Question

For every interval $[a,b]$, there exists a uniform probability density over this interval, which is the constant function $f(x)=\frac{1}{|a-b|}$ for $a < x < b$, and $f(x)=0$ for all other $x$.

Now it's clear that there is no ordinary probability density which is constant on all of $\mathbb{R}$ which also has integral $1$. (The height of such a function would have to be infinitesimally small.)

However, it seems to me as if the Dirac-Delta $\delta_x$ has very similar problems: It is conceptually a function that is infinitely thin but still has integral $1$.

In the Dirac-Delta case, one could solve the issue by generalizing the definition of a function and defining $\delta$ not as a function, but as a Schwartz distribution.

So my question is: Is there some way, for example by generalizing some definition somewhere, to get a function that acts like a uniform probability density over the real numbers, i.e. when multiplied with any other density $f$ the resulting density will be equal to $f$ (up to a constant factor)?

Answer 1

(1) Google the term "improper prior".

(2) Look at the works of Edwin Jaynes.

(3) Maybe even search for "improper prior" in the works of Edwin Jaynes.

My suspicion (really only a suspicion) is that something broader than probability is involved. If you restrict yourself to probability as conceived by Kolmogorov's system of axioms then of course the answer is "there is no ordinary probability density which is constant on all of $\mathbb{R}$ which also has integral 1." But Kolmogorov's axioms are not the last word. Countable, as opposed to finite, additivity has been questioned by de Finetti, maybe Renyi, and others. There's a book called Theory of Charges (by K. P. S. Bhaskara Rao and M. Bhaskara Rao) that deals with finitely additive measures that are not necessarily countably additive, that may be worth checking out. But quite aside from the question of whether Kolmogorov is God's last prophet in the field of probability, I think one might wonder whether probability itself is the last word.

(Admittedly, Jaynes is annoying at times. And sometimes he's immensely entertaining. You can also find his writings on music, including stuff on the thermodynamics of muscles in human fingers as applied to piano technique.)

Later note: I don't know why, but I forgot to mention Sir Harold Jeffreys' book on probability theory. His Ph.D. was in mathematics. For a time he was professor of astronomy. He wrote a lot of stuff about seismology. And perhaps only because of an appendix in one edition of one of his somewhat philosophical books do we know today about Mary Cartwright's proof of the irrationality of $\pi$. Anyway, you'll find a fair amount of stuff on improper priors in that book, and maybe a certain amount of disdain for logical rigor as understood by mathematicians.