Largeness, generic, random points

Question

As presented in Oxtoby's book ( http://link.springer.com/book/10.1007%2F978-1-4615-9964-7 ), there are two notions of largeness for subspace $Y$ of a given space $X$:

1. Topology: $X$ is a topological space, and $Y$ is large if its complement is a countable union of nowhere dense subsets.
2. Measure: $X$ is a probabilistic space, and $Y$ is large if $Y$ has measure $1$.

These two notions do not coincide in general. They are often used to state how a property $P$ is "true", i.e., true on a large set (either topolgy or measure-wise).

It seems, in some cases, that a property $P$ being true on a large set is equivalent to $P$ being true of a single point. So far, I have seen the notions of

1. Generic point in the context of algebraic geometry. For instance, a polynomial vanishes on a generic point iff it vanishes everywhere (see e.g. https://mathoverflow.net/a/92031/59239)
2. Martin-Löf random point: this has a more measure-theoretic flavour, as it can be defined as a point not satisfying any Martin-Löf test on a given (computable) probability space. Yet, here, I'm not sure if a property being true on a martin-löf random point is necessarily true almost everywhere, but my guess is this is true (modulo some constraints on the type of property).

Are these similarities superficial ? Are there other examples of notions of largeness which also admit their own definitions of "generic point" ? Is there a systematic (or natural) association of "def of genericity" with "def of largeness" ?

Thanks

Answer 1

Note that in the Martin-Löf case, if an ML random point has a property of a suitable kind then the set of points with that property has positive measure, not necessarily measure 1. (There's a whole zoo of related notions coming from algorithmic randomness, which mostly have the same basic structure, but where the kind of properties in question vary.) Also related to such notions is the ergodic theoretic notion of generic (a point is generic in this sense if for every continuous function, the ergodic average of a function at that point converges to the integral of the function).

Both kinds of generics and the notions from algorithmic randomness have the same underlying definition---they're the points outside a union of small sets. Their genericity comes from that: if $x$ is not in any small set of a certain kind then whenever $P$ is a set of the right kind and $x\in P$, $P$ must not be small.