The existence of a regular Borel
measure whose support is a given compact Hausdorff space X imposes definite
structures on X, C(X), and C(X)∗. In this paper a necessary and sufficient condition
is given to insure that X is the support of a regular Borel measure. This involves the
intersection number of a collection of open sets in X. Measures which vanish on a
sigma ideal of a sigma field of subsets of X which contains a basis for the
topology of X are also considered. In particular, for a certain class of compact
Hausdorff spacs X, necessary and sufficient conditions are given to insure
the existence of a nonatomic regular Borel measure whose support is X.
The final section of the paper is devoted to a study of normal measures;
i.e., measures which vanish on meager Borel sets. Normal measures on X
are shown to be related to normal measures on the projective resolution of
X.
