[emblaegh]

You first split the sphere into an uncountable number of subsets, then group these into a finite number of subsets, whose measure sum to twice the measure of the original set.

(Un)countability is at the core of most of the counterintuitive results of measure theory, exactly because of the the third property of measure.

[adroniser]

To be clear, the construction given here violates the finite additivity property of measure. It's got nothing to do with the countable/uncountable additivity property.

[emblaegh]

Yeah but this is only possible if the sets in question are uncountably infinite.

[adroniser]

Yes but you are not taking an uncountable union. You are taking a finite union.