We know that a cardinal $k > \omega$ is measurable if there is a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies the following 3 conditions:
1.$<k$ -additive: for every set I of indices with card(I) < k, and for every family of pairwise disjoint sets $z_i$, where $i\in I$, we have $\displaystyle \mu(\bigcup_{i\in I} z_i)=\sum_{i\in I} \mu(z_i)$.
2.$\mu(k)=1$
3.$\mu (s)=0$ for every singleton.
Ok, now what if there exists a cardinal $k>\omega$, and a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies conditions 2 and 3 above, but is only $ <\omega_1$ -additive ? How can we show that this weaker condition still implies the existence of a measurable cardinal?
Thanks a lot!