Question about Royden's proof about measurable set

Question

I am reading the following proof(Lemma 11) by Royden

I would like to know why there is inequality instead of equality in the second last inequaility (the one that I circled and put a question mark beside).

Isn't it that $\Sigma(m^*I_n^\prime+m^*I_n^{\prime\prime}) = \Sigma l(I_n)$?

Note: $m^*$ denotes Lebesgue outer measure

Answer 1

Royden probably copied that proof from somewhere that were dealing with outer measures in general. In general, you only have the $\sigma$-sub-additivity.