We give the general variational form of
\[
lim sup
(
∫
X
e
h
(
x
)
/
t
α
μ
α
(
d
x
)
)
t
α
\limsup (\int _X e^{h(x)/t_{\alpha }}\mu _{\alpha }(dx))^{t_{\alpha }}
\]
for any bounded above Borel measurable function
h
h
on a topological space
X
X
, where
(
μ
α
)
(\mu _{\alpha })
is a net of Borel probability measures on
X
X
, and
(
t
α
)
(t_{\alpha })
a net in
]
0
,
∞
[
]0,\infty [
converging to
0
0
. When
X
X
is normal, we obtain a criterion in order to have a limit in the above expression for all
h
h
continuous bounded, and deduce new criteria of a large deviation principle with not necessarily tight rate function; this allows us to remove the tightness hypothesis in various classical theorems.