Let
(
X
,
Σ
,
μ
)
(X,\Sigma ,\mu )
be a
σ
\sigma
-finite measure space and let
T
:
L
1
(
X
,
Σ
,
μ
)
→
L
1
(
X
,
Σ
,
μ
)
T : L^1(X,\Sigma ,\mu ) \to L^1(X,\Sigma ,\mu )
be a Frobenius-Perron operator. In 1997 Bartoszek and Brown proved that if
T
T
overlaps supports and if there exists
h
∈
L
1
(
X
,
Σ
,
μ
)
h \in L^1(X,\Sigma ,\mu )
,
h
>
0
h > 0
on
X
X
, such that
T
h
=
h
Th = h
, then
T
T
is (strongly) asymptotically stable. In the note we prove that instead of assuming that
h
>
0
h > 0
on
X
X
, it is enough to assume that
h
≥
0
h\geq 0
and
h
≠
0
h\neq 0
. More precisely, we prove that
T
T
is asymptotically stable if and only if
T
T
overlaps supports and there exists
h
∈
L
1
(
X
,
Σ
,
μ
)
h\in L^1(X,\Sigma ,\mu )
,
h
≥
0
h\geq 0
,
h
≠
0
h\neq 0
, such that
T
h
=
h
Th=h
.