Existence of non-atomic probability measure The Question
Let $X$ be a set. Let $\mathcal{F}\subseteq P(X)$ be a $\sigma$-algebra. (Or, if it makes a difference, let $X$ be a topological space and $\mathcal{F}$ the Borel sets.) When can we guarantee the existence of and how can we construct a non-atomic probability measure $\mu$ on $(X,\mathcal{F})$? (In addition to being a measure, we ask that $\mu(X) = 1$ and that if $F\in\mathcal{F}$ and $\mu(F) \neq 0$, there exists a proper measurable subset $E\subsetneq F$ such that $0 < \mu(E) < \mu(F)$.)
Some remarks
The result of Sierpinski guarantees that $\mu$ actually takes on a continuum of different values. So necessarily $\mathcal{F}$ needs to be uncountable and hence $P(X)$ has to be uncountable. This puts a lower bound on how many elements there can be in $X$. (And trivially one sees that if $X$ is finite any probability measure must have atoms.) 
I expect, however, that there may be other necessary conditions for the existence of a non-atomic probability measure. 
On the sufficient side, the only result I am familiar with that explicitly constructs a measure is the various constructions of the Lebesgue measure. This construction makes use of the local structure of Euclidean space and hence also works for, say, topological manifolds. (Okay, there's also the ultrafilter construction for additive measures, but while the constructed measures are probability, they have a lot of atoms.) 
 A: Following Willie Wong, the answer to his question depends on the cardinality of $X$. 
In the case when $card(X)=c$ his question is not uniquely solvable
in the theory $(ZF)\&(DC)$ for $X=[0,1]$ and $\cal{F}=P[0,1]$, where
$(ZF)$ denotes the  Zermelo-Fraenkel set theory and $(DC)$
denotes the axiom of Dependent Choices.
Indeed, on the one hand, in the consistent theory $ZF \&DC \& AD$, where $AD$ denotes an Axiom of Determinacy, the answer to his question is yes, because  Mycielski and Swierczkowski well known result asserts  that every subset of the real axis is Lebesgue measurable. Hence such a measure is exactly Lebesgue measure in $[0.1]$.
On the other hand, in the consistent theory $ZF\& DC \& AC \& \omega_1=2^{\omega}$ by Ulam's well known result  on the powerset of $\omega_1$ (correspondingly, of $2^{\omega}$) we can not define a probability measure which vanishes on singletons.
Since both theories are consistent extensions of the theory $ZF\& DC$ we deduce that Willie Wong's question is not solvable within the theory $ZF\&DC$ for $X=[0,1]$ and $\cal{F}=P[0,1]$.
