Talking about properties of “random” elements

Question

I asked this in MSE but did not get a satisfying answer. I apologize in advance if this is not appropriate for MO.

Suppose that we have some set X and we want to say that a "random" (or generic) element of X has some property, say P.

As far as I understand one way of doing this is to put a probability measure μ on X and show that the elements of X having property P are of μ measure 1.

Of course this will heavily depend on the measure one has. If, for example there is $x\in X$ with property P, and one takes the point measure $μ=δ_x$, then a "random" element will have property $P$. This, being mathematically perfectly fine, intuitively is not as nice since $x$ may be the only element in $X$ with property $P$.

My question is what kind of model would be considered as a "nice" model for a random element of a set $X$? In other words, what type of property of $μ$ would make the sentence "A $μ$ random element of $X$ has property $P$" interesting?

Answer 1

This question is actually much more meaningful than some local wits imagine, so let me answer before it's closed. Indeed, it boils down to finding "nice" or "natural" measures on your state space $X$. Now, in order to talk about these niceties one needs appropriate words - in other words, it only makes sense in the presence of additional structures on your set; then what is natural is usually quite easily seen in terms of these structures. Luckily, sets appearing in mathematical discourse are usually not "naked". If your set is finite, then as a last resort one can always take the uniform measure.

Now, it quite often happens that the raison d'être of a certain measure (or a family of measures, if one talks about a collection of sets) is not a certain a priori naturality, but just the fact that these measures are easier to deal with (which ultimately boils down to the fact that they agree with a certain structure) - this is yet another instance of the famous streetlight effect.

For example, when one says that a certain property $P$ of a natural number is "typical", quite often it means that the asymptotic density of the set of points with this property is 1. The definition of asymptotic density is based on Cesaro averaging - the corresponding sequence of probability measures on $\mathbb Z$ are just the uniform measures on intervals $[1,n]$.

Now, one can ask the same question about a certain property of elements in a countable (infinite) group $G$, which is what do people in the so-called quantitative group theory. Arguing in the same way one could consider averaging with respect to the sequence of balls in the group (of course, before doing that one has to choose a "nice" or "natural" metric, usually this is the word metric of a finite generating set). However, it is much more difficult to deal the ball averaging than to average along the distributions of a random walk on a group (as, for instance, in the latter case the corresponding ergodic theorem is almost obvious, whereas for ball averaging it is a difficult and largely open problem). This is why random walk averaging is nowadays considered as perfectly natural in this situation, although some time ago it looked outright exotic and artificial.

Yet another interesting problem consists in comparing "natural measures" arising from different structures coexisting on the same state space. This problem arises in various areas and is notoriously difficult. The standard answer is easy to guess: coincidence or equivalence of different natural measures means that the corresponding structures agree in a nice way, but usually this is quite hard to establish rigorously. Examples are, for instance, maximal entropy and smooth measures in dynamical systems, Hausdorff, harmonic and invariant measures in fractal analysis or Liouville, Patterson and harmonic measures in the study of negatively curved manifolds, etc.