  This helps, but not much. For there are many measure one subsets of the Cantor space, and we need some non-arbitrary way of selecting a privileged such subset. (The natural option, to take the intersection of all measure one subsets, fails, because the complement of the singleton of any specific sequence is measure one, so for each sequence there is a measure one set which excludes it; therefore the intersection of all measure one sets excludes every sequence, so is the empty set.) The usual response is to take the random sequences to be the intersection of all measure one subsets of the space which have ‘nice’ properties, and to give some principled delimitation of which properties are to count as ‘nice’ and why.