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The probabilistic functor $P$ sends a measurable space $X$ to the space of probability measures on $X$ endowed with $\sigma$-algebra generated by evaluation maps, and measurable maps $f:X\to Y$ to pushforwards of measures $f_*:P(X)\to P(Y)$. There are versions of $P$ on Borel/Polish spaces only (in this case $P(X)$ is endowed with the topology of weak convergence) etc.

The notation $P(X)$ is rather natural, but I have often seen it also used for a very similar powerset functor. I wonder which single-letter notation is the most commonly used for the probabilistic functor. Although this question is not of a technical level, it seems to me that MO community may have better suggestions than MSE one, so I post it here.

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  • $\begingroup$ I don't know, but it has to be a single letter? I'd be inclined to use something like $\text{Pr}(X)$. $\endgroup$
    – Todd Trimble
    Commented Jun 27, 2014 at 11:20
  • $\begingroup$ @ToddTrimble: ideally, yes - and I also would expect so. I've seen the use of $P$, $\mathcal P$ - but they overlap with powerset functor denoted by the same letters (rather than $\mathfrak{Pow}$, for example). Also, $\Delta$ and Giry used $\Pi$. $\endgroup$
    – SBF
    Commented Jun 27, 2014 at 11:35

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In my experience, $P(X)$ or $\mathcal{P}(X)$ is the most common notation.

Yes, they overlap with the notation for "power set", but of course we all know there is no injection from the set (class?) of mathematical concepts to the set of notations. It should be clear from context which is meant. If you need to use both in the same paper, you might consider writing the power set as $2^X$ instead.

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    $\begingroup$ Isn't the set of "mathematical concepts" countable? ;-) Each has a finite description and you could use that description (represented in symbols) as the notation. $\endgroup$
    – Algernon
    Commented Jun 28, 2014 at 7:33
  • $\begingroup$ Thanks for sharing your experience, Nate - perhaps I'll stick to $\mathcal P$ as I used it before. $\endgroup$
    – SBF
    Commented Jun 28, 2014 at 15:10
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I use $\Delta$ both because it makes me think of the "d" at the beginning of "distribution" and because when $X$ is a discrete set of cardinality $\lvert X \rvert = 3$ then $\Delta(X)$ actually is a triangle.

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  • $\begingroup$ Interesting, I haven't ever heard of either of these two reasonings :) $\endgroup$
    – SBF
    Commented Jun 28, 2014 at 15:10

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