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 $X$:
- Topology: $X$ is a topological space, and $Y$ is large if its complement is a countable union of nowhere dense subsets.
- 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
- 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)
- 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" ?