  In large deviations theory one usually has a family of probability measures
in some space $X$ with a topology. In this paper we study what kind of large
deviation results can be established when no topology is assumed on the
underlying measure space $X$. The main tool is duality theory of risk measures.
We prove that a monetary risk measures $\phi$ satisfies the Laplace principle
$\phi(f)={\rm ess.sup}_X(f-I)$ for some measurable function $I\colon
X\to[0,\infty]$ if and only $\phi$ is maxitive and continuous from below. In
the special case of the asymptotic entropy $\psi(f)=\lim_{n\to\infty}
\tfrac{1}{n}\log \int e^{n f} d\nu_n$, this result can been understood as a
non-topological analogue of the Varadhan's integral lemma. In the spirit of
Bryc's theorem, we study non-topological analogues of exponential tightness
which ensure the Laplace principle characterised in the main result. The theory
is applied to obtain non-topological large deviation results for sequences of
sublinear expectations.

Probability