Inequivalent definitions of the support of a Borel measure All definitions appearing in this question apply to Borel measures.
Many sources (including Wikipedia and Folland) define the support of a Borel measure to be the set of points whose open neighborhoods all have positive measure, or equivalently


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*the intersection of the closed sets of full measure, or

*the complement of the union of the open null sets.


Some other sources (including Bogachev) declare a set to be the support if and only if it is a minimal closed set of full measure. This definition implicitly acknowledges that such a set may not exist.
Finally, there are some (including Kallenberg) that declare the support to be the smallest closed set of full measure. Is this notion well-defined? If not, what is a (preferably explicit and self-contained with all details worked out) example of a Borel measure for which there does not exist a smallest closed set of full measure?
Edit: According to the responses to this question, the third definition is ill-defined in the absence of regularity assumptions. This seems to be a (relatively minor) mistake in Kallenberg's book (and in other places where this definition is used) or at least a case of implicit assumptions.
 A: The minimal (or smallest) closed set of full measure need not always be well-defined.
The "example" (which we cannot show to exist in ZFC) is a discrete space $X$ with a probability measure $\mu$ on $\mathcal{P}(X)$ (look up real-valued measurable cardinals) that is $0$ on all singletons: for every closed set $C$ with $\mu(C)=1$, then if $x \in C$, $\mu(C\setminus\{x\}) =1$ too and this set is closed and strictly smaller. So only the empty set is minimal.
There are no points all of whose neighbourhoods are of positive measure. If course this measure is not tight, as the only compact sets are finite and thus measure $0$. So the intersection of all closed sets of full measure is $\emptyset$ which does not have full measure, and this is indeed the complement of the union all open null sets. So in that sense it's defined, but it's "useless".
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Added: Another classical, better, somewhat related example (in ZFC !) is the $0$-$1$-probability measure on $\omega_1$, which is explained here, the so-called Dieudonné measure (as Halmos calls it in in Measure Theory): 
The set $X$ is just the ordinal number $\omega_1$ in its order topology (Munkres calls this space $W$, others call it $\Omega$) and a measure $\mu$ is defined on its Borel sets by $\mu(A) = 1$ iff there is a closed and unbounded set (a "club") that is a subset of $A$; otherwise $\mu(A) = 0$. As clubs are closed under countable intersections, this does define a somewhat unintuitive Borel measure on $X$, with the properties that $\mu(X) =1$, $\mu(\{x\}) = 0$ for all $x \in X$ (a singleton is not unbounded) and even $\mu(K) = 0$ for all compact subsets of $X$ (as these are bounded above and so don't contain a club). If $C$ is a closed set of measure $1$ (so it's actually a club) it will contain many isolated points $p$ (all successor ordinals are isolated in $X$) and then $C\setminus \{p\}$ is a strictly smaller closed set of full measure. So there cannot be a minimal closed set of full measure. Of course the measure is not inner regular for compact sets, but it is inner regular for closed sets, almost by definition. Ulam already showed that we cannot extend this measure to all subsets of $\omega_1$, but on the Borel sets it's well-defined.
If a probability Borel measure is tight (inner regular for compact sets) then defining $$N = \bigcup \{U| U \text{ open }, \mu(U)=0\}$$ the set $N$ has measure $0$: any compact $K \subseteq N$ is covered by finitely many open null sets, so has measure $0$ too and so $\mu(N) = \sup \{\mu(K): K \subseteq N, K \text{ compact }\} = 0$ and in that case the support 
$$X\setminus \bigcup\{U: U \text{ open }, \mu(U)=0\}=
\bigcap \{ C: C \text{ closed }, \mu(C)=1\}$$ has full measure and is indeed minimal with that property in that case. 
So the “problem” is solved by considering suitably regular Borel measures. That’s why in analysis applications only Radon or tight measures are considered. See also the Handbook of Set-theoretic Topology (1984), one of my "bibles", in the chapter on Borel measures, for more discussion of such issues.
