We infer from MCT that $v$ is a measure (Radon-Nikodym)

Question

This is from Royden text. He say that "We infer from the linearity of integration and the MCT that $v$ is a measure .." I understand the linearity of integration $\implies$ $v$ is measure. But, I am not sure how MCT is used in this case. Could you give some hint?

Answer 1

If $\{E_n\}$ is a disjoint sequence in the sigma algebra with union $E$ then $fI_E=\sum_{n=1}^{\infty} fI_{E_n}$. Let $f_k =\sum_{n=1}^{k} fI_{E_n}$ and apply MCT to this sequence.