1) Consider a measurable space $(X, \mathcal A)$. Let $\delta_x$ denote the Dirac measure concentrated at $x \in X$. Then I found a very primitive counterexample to the implication $\delta_x = \delta_y \Rightarrow x=y$: just take $\mathcal A = \{\emptyset, \{x,y\}\}$ on $X= \{x,y\}$. What assumptions on $\mathcal A$ assure that the implication always holds?
2) Given a sequence $(x_n)$ of real numbers, we can define a measure $\mu$ given as the sum $\mu = \sum_{n \geq 1} \delta_{x_n}$ of delta measures. Is my understanding correct that this gives a sigma-finite measure on the Borel sigma-algebra on $\mathbb R$ if every real number occurs in the sequence $(x_n)$ only finitely often?
3) More informal: Is there a notion which characterizes countable linear combinations of Dirac-measures? (I don't know exactly what I am expecting here but these kind of measures seem very canonical to me)
Ad 3) Those measures for which there is a countable set $M $ with $\mu (X\setminus M)=0$.
Ad 1) It certainly suffices if each singleton is measurable.
Ad 2) Yes, this is true.