Let $\omega$ be a positive linear functional on $M$ which is a Von Neumann Algebra. Suppose $\omega$ is completely additive (i.e. $\omega$ applied to a strongly convergent sum of mutually orthogonal projections is the same as the convergent complex valued sum of $\omega$ applied to the projections). Then supposedly, $\omega$ is the pointwise convergent sum:
$\sum\limits_{i=0}^\infty \omega_{\xi_n} \text{ where } \omega_{\xi_n}(x)=\langle\xi_n, x(\xi_n)\rangle \qquad\forall x \in M$
Please prove this for me, but beware to not use the Cauchy Schwarz inequality in an incorrect way. The reason I am without a proof is that the proof in a document I am reading is actually incorrect, and uses Cauchy Schwarz incorrectly.