The following point has been addressed in the significant paper "order ideal in C*-algebras and its dual (By E. Effros - Lemma 4.1). The reference is Dixmier's book (1957 French). Unfortunately, I have no access.
1- Let $\phi$ be a normal positive linear functional on a von Neumann algebra and assume that $\psi$ is a normal positive functional on $M$ such that support$(\phi)$ mojorizes support$(\psi)$. Let us assume $(\pi,\zeta,H_{\zeta})$ be the cyclic normal representation corresponding to $\phi$. Then (?) there is a vector $\eta\in \overline{\pi(M)'\zeta}$ with $\psi=\langle\pi(\cdot)\eta,\eta\rangle$.
2- In this paper, again in the Theorem 4.4 a fact form Dixmier's book is addressed: Let $0\leq\psi\leq \phi$ in $M_*$. Assume $(\pi,\zeta,H_{\zeta})$ be the cyclic normal representation corresponding to $\phi$. Then (?!) there is a vector $\eta\in H_{\zeta}$ with $\psi=\langle\pi(\cdot)\zeta ,\eta\rangle$.