In the French Wikipedia article https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Radon-Nikodym-Lebesgue, the definition of a being supported by a subset is :
$\rho$ is said to be supported by $E\in\mathcal A$ (or be concentrated on $E$) if for all $A\in\mathcal A$ there is $\rho(A) = \rho(A \cap E)$.
I did a demonstration (which I could write if necessary) that this is equivalent to $\rho(E) = 1$. As it is much simpler than the definition proposed, I suppose my demonstration is flawed. So is my equivalence false, and if this is the case, could you provide a counter-example ?