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Let there be a $\text{probability space}$ $(\Omega , 2^\Omega , \mathbb{P})$, where $\mathbb{P}$ is mapping $\mathbb{P}: 2^\Omega \mapsto [0,1]$ , that is, $\mathbb{P}$ is called a probability measure (sometimes "a set function", but never "a distribution") on $2^\Omega$. And $X$ is a random variable $X:\Omega\to\mathbb R$.

Now, the distribution of the random variable $X:\Omega\to\mathbb R$ is the unique probability measure $P$ on $\mathcal B(\mathbb R)$ defined by $$P(B):=\mathbb{P}(X\in B)$$ for every $B$ in $\mathcal B(\mathbb R)$, where $[X\in B]=X^{-1}(B)=\{\omega\in\Omega\mid X(\omega)\in B\}$.

So, the question is:

Why call $P$ a probability distribution? Is there some logic behind the choice of this technical term for $P$? All I see is another function $P$, existence of which was induced by existence of $\mathbb{P}$ and $X$. Well, yes there is, probably, a need to distinguish between measure $\mathbb{P}$ and measure $P$, but to make up new term (distribution), for the sole purpose of differentiating them in talk, is too much.

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    It tells you how much weight is placed on a given set. In general, it tells you how the weight is distributed. – 2017-02-09
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    Sorry, but why the same can't be said about the $\mathbb{P}$? I just don't see the "weight" thing. Can you elaborate on the subject. thanks. – 2017-02-09
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    If you repeatedly run an experiment with your random variable $X$, then $P$ models the *distribution* of results (i.e. predicts how much of each result we should expect). That's what I think they're getting at – 2017-02-09
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    If I run experiment without the notion $X$, that is not trying to link objects of of the space $\Omega$ with numbers $\mathbb{R}$, then $\mathbb{P}$ too can show the distribution of the results. Can it? – 2017-02-09
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    I guess in principle, yes, but firstly, it has nothing to do with $X$, and secondly, as far as I know, in probability theory you usually do not really care about the space $\Omega$ itself beyond the fact that it exists, so $\Bbb P$ itself is of little interest. It's a mostly auxiliary object. – 2017-02-09
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    But the point is that "random variables" are *made* for modeling such experiments. A probability measure, in and of itself, doesn't suggest this as strongly. – 2017-02-09
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    Ok, If $\Omega$ is an auxilary thing and $\mathbb{P}$ too, then why not call $P$ measure, since absence of $\mathbb{P}$ frees the term "measure" for another use? – 2017-02-09
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    @coobit I think you have the chronology backwards, at least in some sense. The idea of distribution functions to describe a statistical law existed decades before Komolgorov's recasting of probability theory in terms of measure spaces in 1933. – 2017-02-09
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    Well, $P$ is a specific measure induced by random variable $X$. You could call it "a pushforward measure induced by $X$" or "a distribution of $X$". It is just a name. – 2017-02-09
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    @coobit I think the point is that the word "distribution" characterizes the relationship between the measure and the random variable in question. Any probability measure *can* be the distribution of *some* random variable, but if we're not considering a random variable, then we're not thinking about the "distribution" of anything. – 2017-02-09
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    @Omnomnomnom, "Any probability measure can be the distribution of some random variable..." My head hurts. Can you make into a full answer? – 2017-02-09
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    @Erick Wong, ok if the $P$ existed long before "measure theory" and before any use of "measure", then why not call $P$ a "probability" or "probability function" or anthing else? Then after measure was "invented" why not call $\mathbb{P}$ a distribution on $\Omega$? – 2017-02-09
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    @coobit Classically, $P$ existed in the form of the cumulative distribution $F(t) = \mathbb P(X < t)$. It certainly makes sense to call this a distribution rather than a measure, and indeed it makes sense without having the concept of measure. Once you have such a distribution then countable additivity extends this to a measure on Borel sets, perhaps it is natural to name this extension a distribution also. To your question why not call $\mathbb P$ a distribution: what variable is it the distribution of?? – 2017-02-09

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