This note deals with
(
M
,
∗
)
({\mathbf {M}},\ast )
functions for various families
M
{\mathbf {M}}
. It is shown that if
M
{\mathbf {M}}
is the family of Borel sets of additive class
α
\alpha
on a metric space
X
X
, then
(
M
,
∗
)
({\mathbf {M}},\ast )
functions are just the functions of the form
sup
y
g
(
x
,
y
)
{\sup _y}g(x,y)
where
g
:
X
×
R
→
R
g:X \times R \to R
is continuous in
y
y
and of class
α
\alpha
in
x
x
. If
M
{\mathbf {M}}
is the class of analytic sets in a Polish space
X
X
, then the
(
M
,
∗
)
({\mathbf {M}},\ast )
functions dominating a Borel function are just the functions
sup
y
g
(
x
,
y
)
{\sup _y}g(x,y)
where
g
g
is a real valued Borel function on
X
2
{X^2}
. It is also shown that there is an
A
A
-function
f
f
defined on an uncountable Polish space
X
X
and an analytic subset
C
C
of the real line such that
f
−
1
(
C
)
∉
{f^{ - 1}}(C) \notin
the
σ
\sigma
-algebra generated by the analytic sets on
X
X
.