Characterizing measure spaces Let $(\Omega, B, \mu)$ be a measure space. How can we characterize the space $(\Omega, B, \mu)$ so that  


*

*the counting measure on $B$ is absolutely continuous with respect to $\mu$?   

*the Dirac measure on $B$ is absolutely continuous with respect to $\mu$? 


Edit: What I mean by the Dirac measure on $B$ is the following: Let $x\in \Omega$. Then the Dirac measure at $x$ assigns $1$ to a set in $B$ that contains $x$ and $0$ to a set that does not contain $x$.
 A: *

*Counting measure is usually defined on the $\sigma$-algebra of all subsets. So let $\mu$ be another measure defined on all subsets. Counting measure is absolutely continuous with respect to $\mu$ if no $\mu$-zero set has positive counting measure. Every nonempty set has positive counting measure. So $\mu$ needs to put positve measure on every nonempty-set, in particular on every singleton. The measure on all singletons determines the measure on all countable sets and since uncountable sums of positve real numbers are always infinite, every such measure is determined by having a positive value at each singleton. So for $\mu$ there exists a function $f:\Omega\to\mathbb{R}\cup\{\infty\}$ with strictly positive values such that $\mu(A)=\sum_{\omega\in A}f(\omega)$ and this property characterizes the measures you are looking for. 

*Dirac measures are bit more delicate since they can be defined on any $\sigma$-algebra. $\delta_x$ is absolutely continuous with respect to $\mu$ if $\mu(A)>0$ for every measurable set containing $x$. If the $\sigma$-algebra contains a smallest set $S$ containing $x$, which is true if it is countably generated or the powerset, then we just need $\mu(S)>0$.
