The set "X" will be non-measurable for any countably additive probability measure on "S": if "X" has zero measure, countable additivity would imply that the whole circle has zero measure.
The set "X" will be non-measurable if the measure is assumed to be a countably additive probability measure. Namely, if "X" has zero measure, countable additivity would imply that the whole circle has zero measure.