Let X be a complete, separable metric space, and
a family of probability measures on the Borel subsets of X. We say that
obeys the large deviation principle (LDP) with a rate function I( · ) if there exists a function I( · ) from X into [0, ∞] satisfying:
(i) 0 ≦ I(x) ≦ ∞ for all x ∊ X,
(ii) I( · ) is lower semicontinuous,
(iii) for each 1 < ∞ the set {x:I(x) ≦ 1} is compact set in X,
(iv) for each closed set C ⊂ X
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00000201/resource/name/S0008414X00000201_eq1.gif?pub-status=live)
(v) for each open set U ⊂ X
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00000201/resource/name/S0008414X00000201_eq2.gif?pub-status=live)
It is easy to see that if A is a Borel set such that
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00000201/resource/name/S0008414X00000201_eq3.gif?pub-status=live)
then
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00000201/resource/name/S0008414X00000201_eq4.gif?pub-status=live)
where A0 and Ā are respectively the interior and the closure of the Borel set A.