I understand the proof for the subadditivity property of the outer measure (using the epsilon/2^n method), but I am not quite clear on the proof for the sigma-additivity property of measures. Most sources I have read either leave it an exercise or just state it outright.
From what I gather, they essentially try to show that a measure is also *super*additive (the reverse of subadditive) which means it must be sigma-additive. However, I'm a bit confused as to how they do this.
Would anyone be kind enough to give a simple proof about how this could be done?