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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?

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  • $\begingroup$ Isn't the second possibility just a special case of the first, when $X$ has some topological structure? $\endgroup$ – grndl Mar 20 '17 at 17:38
  • $\begingroup$ @aduh, sure, but the emphasis is on $\mathbb{R}$ in this question. I want to know whether it's best to focus on the $\sigma$-algebra of Lebesgue-measurable functions, as compared to the $\sigma$-algebra generated by the topology. You're right that the latter is basically a special case of the former, but this doesn't really answer the actual question. $\endgroup$ – goblin Mar 20 '17 at 17:40
  • $\begingroup$ It depends what the distribution looks like. In principle you could use whatever $\sigma$-algebra you like, but generally you want your $\sigma$-algebra to contain the Borel sets if only for convenience. $\endgroup$ – Ian Mar 20 '17 at 17:44
  • $\begingroup$ @goblin Ah I see. If you look around this site you can find examples of continuous functions that are not measurable when their range space is equipped with the Lebesgue measurable sets. On the other hand, all continuous functions are measurable for the Borel sigma-algebra. See this, for example. Does that help at all? $\endgroup$ – grndl Mar 20 '17 at 17:55
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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.

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  • $\begingroup$ Can you elaborate on why this is a good definition? I should add that I don't particularly like the tone of the opening sentence. $\endgroup$ – goblin Mar 20 '17 at 17:52
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    $\begingroup$ @goblin : I'm sorry you don't like it, but your question is in some ways rather strange, saying things like "as far as I can tell, the phrase probability distribution is never really defined" when it's in all standard textbooks (any book that introduces a measure-theoretic theory of probability). $\endgroup$ – Michael Hardy Mar 20 '17 at 17:58
  • $\begingroup$ I'm under the impression that it's the phrase "probability measure", not "probability distribution" that's actually defined. Can you point me to a reference that defines probability distribution as you're defining it, preferably one with some further justification as to why this is the "correct" $\sigma$-algebra to focus on? I should emphasize that my question is really: what should "not-necessarily deterministic point of the real line mean"? Best not to fixate too much on the phrase "probability distribution." $\endgroup$ – goblin Mar 20 '17 at 18:07
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    $\begingroup$ @goblin : A probability distribution on $\mathbb R$ is a probability measure on the set of Borel subsets of $\mathbb R$. The reason the Borel subsets are the right $\sigma$-algebra is that $(1)$ we want the distribution to determine the probability assigned to $(-\infty,x]$ for every $x\in\mathbb R$ and $(2)$ we don't want to go beyond Borel sets in the general definition for the reason stated in my answer. From page 12 of Fristedt & Gray's A Modern Approach to Probability Theory: "Let $X$ be a random variable from a probability space $(\Omega,\mathcal F, P) \text{ to a }\ldots\qquad$ $\endgroup$ – Michael Hardy Mar 20 '17 at 18:51
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    $\begingroup$ $\ldots\,$measurable space $(\Psi,\mathcal G).$ For $B\in\mathcal G$ let $Q(B) = P(X^{-1}(B)).$ Then $(\Psi,\mathcal G, Q)$ is a probability space. The probability measure $Q$ is called the distribution of $X$ and is said to be induced by $X$ (or by $X$ from $P$)." $\qquad$ $\endgroup$ – Michael Hardy Mar 20 '17 at 18:55

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