Definition 1.1. The -rate function associated with a rate function is defined as

Proposition 1.2 (Uniqueness). A family of probability measures on a regular topological space can have at most one associated rate function.

Proof. Suppose and are two rate functions associated with the LDP of , and assume there exists a point such that . By lower semicontinuity, there exists an open neighborhood of such that , where is the -rate function defined in Definition 1.1. By regularity, we can further assume . Consequently,

which leads to a contradiction as .⁠ 

Remark 1.3.

1. All metric spaces are regular Hausdorff spaces.
2. If is regular and locally compact, the LDP condition in Proposition 1.2 can be relaxed to a weak LDP.

Proposition 1.4 (Existence). Let be a topological base for space . For every , define

and

If , then satisfies the weak LDP with the rate function .

Proof. That and are indeed rate functions follows naturally from their definitions.

To establish the lower bound of the LDP, observe that for every open set and any , there exists an such that . This implies

from which the desired lower bound follows.

Regarding the upper bound, for any compact set and every point , we can find a neighborhood satisfying . Since is compact, it admits a finite subcover ; thus,

Taking completes the proof.⁠ 

Remark 1.5. Proposition 1.4 holds for a parametrized family for any given if we define:

This is useful for describing processes such as Markov chains conditioned on the initial state .

Proposition 1.6. Suppose that satisfies the LDP in a topological space with rate function . Then, for any topological base of , , where these functions are defined as in Proposition 1.4.

Proof. Suppose for some . By the lower semicontinuity of and the regularity of , for every , there exists an open neighborhood of such that

Conversely, by the definition of the LDP, for every , there exists an open neighborhood of such that

Combining these inequalities yields the desired result.⁠