In Analytic mappings on hyperfinite sets [Proc. Amer. Math. Soc. 2 (1993), 587-596] Henson and Ross asked for what hyperfinite sets
S
S
and
T
T
does there exists a bijection
f
f
from
S
S
onto
T
T
whose graph is a projective subset of
S
×
T
S \times T
? In particular, when is there a
Π
1
1
\Pi _1^1
bijection from
S
S
onto
T
T
? In this paper we prove that given an internal, bounded measure
μ
\mu
, any
Π
1
1
\Pi _1^1
function is
L
(
μ
)
L(\mu )
a.e. equal to an internal function, where
L
(
μ
)
L(\mu )
is the Loeb measure associated with
μ
\mu
. It follows that if two
Π
1
1
\Pi _1^1
subsets
S
S
and
T
T
of a hyperfinite set
X
X
are
Π
1
1
\Pi _1^1
bijective, then
S
S
and
T
T
have the same measure for every uniformly distributed counting measure
μ
\mu
. When
S
S
and
T
T
are internal it turns out that any
Π
1
1
\Pi _1^1
bijection between them must already be Borel. We also prove that if a
Π
1
1
\Pi _1^1
graph in the product of two hyperfinite sets
X
X
and
Y
Y
is universal for all internal subsets of
Y
Y
, then
|
X
|
⩾
2
|
Y
|
|X| \geqslant {2^{|Y|}}
, which is a partial answer to Henson and Ross’s Problem 1.5. At the end we prove some standard results about the projections and a structure of co-proper
K
K
-analytic subsets of the product of two completely regular Hausdorff topological spaces with open vertical sections. We were able to prove the above results by revealing the structure of
Π
1
1
\Pi _1^1
subsets of the products
X
×
Y
X \times Y
of two internal sets
X
X
and
Y
Y
, all of whose
Y
Y
-sections are
Σ
1
0
(
κ
)
\Sigma _1^0(\kappa )
sets.