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GC Rota, possibly in "Twelve problems in probability theory no one likes to bring up" (verify?), wrote that the point-free definition of a random variable goes in the opposite direction from that normally taught: it is a functor from a Borel algebra on the reals (or complex) numbers to the sample space.

I've been looking but cannot source the original definition.

What properties of a Borel algebra, resp sample space, does such a functor preserve and reflect?

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up vote 3 down vote accepted

Rota used a lot of hyperbole, and "no one likes to bring up" is exaggerated in reference to some of those twelve problems. While Rota was writing this in 1998, he and I exchanged probably a couple of hundred emails about it. I convinced him to use the term "atomless" instead of "non-atomic" since the latter terms seems as if it ought to mean "not atomic", but that's not what it means. I also suggested that rather than merely stating the definitions, he should be completely explicit about the fact that the mapping goes in the opposite direction from what is conventional.

Normally a real random variable is defined as a measurable function $X:\Omega\to\mathbb{R}$, where $\Omega$ is the underlying set of a probability space. For every Borel-measurable subset $S$ of $\mathbb{R}$, you have a measurable set $\{\omega\in\Omega : X(\omega)\in S\}$. This mapping takes subsets of $\mathbb{R}$ to subsets of $\Omega$, rather than points of $\Omega$ to points of $\mathbb{R}$. That's the "opposite direction" aspect of it. But in pointless probability, you never consider points of $\Omega$, or consider $\Omega$. Rather, you have a Boolean algebra of propositions to which you want to assign probabilities. Via "Stone's duality", you can identify each such proposition with a clopen subset of the Stone space of the Boolean algebra.

The Stone space of a Boolean algebra $A$ is the space of Boolean homomorphisms from $A$ into $\{0,1\}$ with the topology of pointwise convergence of nets of such homomorphisms. Every such space is a totally disconnected compact Hausdorff space. Every totally disconnected compact Hausdorff space arises is the Stone space of the Boolean algebra of all of its clopen subsets.

In effect, instead of measurable subsets of $\Omega$, you've got clopen subsets of a Stone space, and you never assign a probability to an individual point in that space except when the set containing just that one point is clopen. Hence this is "pointless".

I don't actually know the original source of "pointless probability theory" (so this doesn't answer that question). The stuff on Boolean algebras and Stone spaces can be found in Halmos' Lectures on Boolean Algebras, which I think is in print again. Halmos never uses the term "Stone space"; rather he defines a "Boolean space" as a totally disconnected compact Hausdorff space. Halmos proves all the basic facts of "Stone's duality" between the category of Boolean algebras and Boolean homomorphisms, and that of Boolean spaces and continuous functions. The two categories are essentially opposites of each other, so there's the reversal of directions again.

Here's the finitary version of the duality: Suppose you have a mapping $f:A\to B$ where $A$, $B$ are finite sets. This is the continuous function from one Boolean space to another. There's a corresponding mapping $g:\mathcal{P}(B)\to\mathcal{P}(A)$ (except that in the general case instead of all subsets of $B$ and of $A$, you have all clopen subsets and you only consider continuous $f$) defined by $g(B_1) = \{x\in A : f(x) \in B_1\}$. $g$ will always be a Boolean homomorphism. In the finite case, that just means it preserves unions, intersections, and complements. In the general case, instead of union, you have the interior of the closure of the union---the smallest clopen set that includes the sets whose "union" is taken. You can show that either of $f$, $g$ is one-to-one if and only if the other is onto.

Rota said he intended to expand his "Twelve Problems" paper to about twice the length it had when he presented it as his "Fubini Lectures" in Torino. But he died suddenly in April 1999.

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The stone space of the events also has a nice interpretation with points being maximal consistent descriptions of the world. There is a kind of survey of pointless probability from 1955: J. Łoś: On the axiomatic treatment of probability. –  Michael Greinecker Jun 27 '12 at 6:10
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For any function $f : X \to Y$ between two sets whatsoever and any subset $S$ of $Y$ we can define the inverse image $$f^{-1}(S) = \{ x \in X | \exists y \in S : f(x) = y \}.$$

This defines an order-preserving map $f^{-1} : 2^Y \to 2^X$. Recalling that a poset (such as $2^X$) is a category in which $a \ge b$ if and only if there is a single arrow $a \to b$ (and otherwise there are no arrows), order-preserving maps are precisely the functors between posets, so $f^{-1}$ is a functor.

I am not sure what you mean by "properties" so I am going to assume you actually mean "limits and colimits." As a functor, $f^{-1}$ is left adjoint to the direct image $$f(S) = \{ y \in Y | \exists x \in S : f(x) = y \}.$$

That is, we have $$f^{-1}(S) \supseteq T \Leftrightarrow S \supseteq f(T).$$

It follows that $f^{-1}$ preserves arbitrary colimits. This is just a fancy way of saying that $f^{-1}$ preserves arbitrary intersections. (Note that $f$ doesn't.)

$f^{-1}$ is also right adjoint, but to a slightly weirder functor, namely $$g(S) = \{ y \in Y | \exists x \in S : f(x) = y \text{ and } \forall x \not \in S : f(x) \neq y \}.$$

It follows that $f^{-1}$ preserves arbitrary limits. This is just a fancy way of saying that $f^{-1}$ preserves arbitrary unions.

Of course you can easily verify both of these properties by hand without using any categorical machinery. You can also verify by hand that $f^{-1}$ preserves complements. Is this the kind of thing you're asking about?

I don't think $f^{-1}$ reflects either limits or colimits in general.

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I'm trying to understand, for example "order-preserving maps are precisely the functors between posets" - but posets are objects while functors act between categories? Are you definining each poset to be a category? –  alancalvitti Aug 15 '12 at 3:30
I remember reading that some categorical properties are not expressible as limits, but whether that was in book by Lawvere or Herrlich or someone else I don't rem. –  alancalvitti Aug 15 '12 at 3:31
@alan: each poset is a category. The objects are the elements of the poset and the morphisms come from the order relation as follows: $a \le b$ means that there is exactly one morphism $a \to b$. –  Qiaochu Yuan Aug 15 '12 at 3:35
@alan: well, every categorical construction is morally a limit or a colimit, it's just that the category it's a limit or colimit in may not be the one you started with. For example the tensor product of two vector spaces $V, W$ is not a limit or a colimit in $\text{Vect}$, but it still has a universal property and as such is still expressible as an initial object in a certain category (namely the category of vector spaces $U$ together with bilinear maps $V \times W \to U$). –  Qiaochu Yuan Aug 15 '12 at 3:36
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