Reference request: generalized randomness

Question

There is a plethora of notions of randomness for Cantor space ($\phantom{}^\omega 2$) (Schnorr randomness, Martin-Löf randomness, weak 2-randomness, the various forms of higher randomness such as $\Delta^1_1$ randomness, and so on). See, for example, André Nies, Computability and Randomness (Oxford: Oxford University Press, 2009), Rodney Downey and Denis Hirschfeldt, Algorithmic Randomness and Complexity (Berlin: Springer, 2010), or any number of other texts. And these can be extended in obvious ways to $\mathbb{R}$, $\phantom{}^\omega \omega$, and some other Polish spaces.

I have been unable, however, to find any discussions of how to extend notions of randomness to $\phantom{}^\alpha 2$ for $\alpha > \omega$, either for $\alpha$ a large countable ordinal or (more ambitiously) where $\vert \alpha \vert > \beth_1$. Any pointers to work on this question would be greatly appreciated.

Answer 1

There have been proposals to generalize the notion of "measure zero sets" to "higher reals", that is, to subsets of ${}^\kappa 2$ for uncountable cardinals $\kappa$, typicallyoften inaccessible or weakly compact. (EDIT: sometimes also singular strong limits)

• Shelah: A parallel to the null ideal for inaccessible $\lambda$, also here

• Friedman-Laguzzi: A null ideal for inaccessibles.

Once you have the notion of a "higher null set", you can define "higher random" to mean "not an element of this or that null set" (for example: definable null set, etc)

(I prefer the adjective "higher" over "generalized", as it is more specific.)