We can think of $\mathbb{R}$ as having deterministic points, like $0$ and $\pi$, and "not-necessarily deterministic" points, like the $(0,1)$-normal distribution and the $(\pi,1)$ normal distribution. So basically, when I say "not-necessarily deterministic point," what I really mean is "probability distribution." Unfortunately, as far as I can tell, the phrase probability distribution is never really defined, so here's my best guess at the correct way of defining it:
Let $(X,\mathcal{M},\mu)$ denote a measure space. Then a probability distribution on this space is a probability measure $\mathcal{M} \rightarrow [0,1]$.
If this is a good definition, then it suggests that the term "not-necessarily deterministic point" of $\mathbb{R}$ ought to mean a probability measure on the Lebesgue-measureable subsets of $\mathbb{R}$. However, maybe it's not such a good definition. Another possibility is:
Let $X$ denote a topological space. Then a probability distribution on this space is a probability measure $\mathcal{B}_X \rightarrow [0,1]$, where $\mathcal{B}_X$ is the Borel sigma algebra of $X$.
If this is a good definition, then it suggests that the term "not-necessarily deterministic point" of $\mathbb{R}$ ought to mean a probability measure on the Borel subsets of $\mathbb{R}$.
My question is therefore:
Question. Is there a consensus on what the phrase "probability distribution of real numbers" should mean? If so, does it mean one of the above two possibilities, and which one? If not, are there nonetheless technical reasons to prefer one of them over another?
That's a really large number of words used just to ask what the standard definition of "probability distribution on $\mathbb R$" is.
In standard usage it is a probability measure on the Borel sets of reals.
The difference between Borel sets and Lebesgue-measurable sets is that if $A\subseteq B\subseteq C$ and $A$ and $C$ are Borel sets with equal measure, then $B$ is a Lebesgue-measurable set with that same measure. However, when a probability measure on Borel sets thus extended by squeezing, the resulting additional measurable sets do not generally coincide with Lebesgue-measurable sets.
It can be shown that if two probability measures on Borel sets of reals both assign the same probabilty to the set $(-\infty,x]$ for every $x\in\mathbb R$, then they agree on all Borel sets. That is why specifying the cumulative probability distribution function is enough to say which probability distribution you're talking about.