Let be $\Omega$ a compact metric space, $\mathcal{B}(\Omega)$ the $\sigma$-algebra of Borelian sets of $\Omega$ and $\mathcal{M}_1(\Omega)$ the set of all probabilities defined on $\mathcal{B}(\Omega)$.

Suppose that $\lambda,\mu\in\mathcal{M}_1(\Omega)$ are extremal points (in the sense of convex combinations) and there is a real number $c$ such that $$ \lambda(A)\leq c\mu(A)\qquad\text{and}\qquad \mu(A)\leq c \lambda(A)\qquad $$ for all $A\in\mathcal{B}(\Omega)$.

Is it true that $\mu=\lambda$ ?

I have proved the above equality in particular cases: 1) $\Omega$ is discrete; 2) $\Omega$ is some subset of $\mathbb{R}$ (the compacity it was not necessary here).