Suppose that we have a probability measure $P$ on space $(S,\mathcal{B}(S))$. There is the statement that says, if $S$ is separable then the support of $P$ always exists. Whereas in the case where $S$ is not separable the support of $P$ may not exist.
The support of the probability measure $P$ is defined as
$$supp(P)= \cap\left \{ \bar{A}\in \mathcal{B}(S)|P (\bar{A}^{c})=0 \right \}$$
Could someone elaborate an example of such a case, where we have a non-separable space with a support that doesn't exist??
A standard example is the so-called Dieudonné measure on $[0,\omega_1)$, where $\omega_1$ is the first uncountable ordinal. A complete description of this measure can be found in this paper or in Example 4.33 [p. 92] in Wise & Hall, Counterexamples in Probability & Real Analysis.