There have been proposals to generated the notion of "measure zero sets" to "higher reals", that is, to subsets of ${}^\kappa 2$ for uncountable cardinals $\kappa$, typically inaccessible or weakly compact:

• 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.)

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.)