A completely regular, Hausdorff space
X
X
is called a Măík space if every Baire measure on
X
X
admits an extension to a regular Borel measure. We answer the questions about Măík spaces asked by Wheeler [29] and study their topological properties. In particular, we give examples of the following spaces: A locally compact, measure compact space which is not weakly Bairedominated; i.e., it has a sequence
F
n
↓
∅
{F_n} \downarrow \emptyset
of regular closed sets such that
∩
n
∈
ω
B
n
≠
∅
{ \cap _{n \in \omega }}{B_n} \ne \emptyset
whenever
B
n
{B_n}
’s are Baire sets with
F
n
⊂
B
n
{F_n} \subset {B_n}
; a countably paracompact, non-Măík space; a locally compact, non-Măík space
X
X
such that the absolute
E
(
X
)
E(X)
is a Măík space; and a locally compact, Măík space
X
X
for which
E
(
X
)
E(X)
is not. It is also proved that Michael’s product space is not weakly Baire-dominated.