A Course on Large Deviations with an Introduction to Gibbs Measures.

2.3. Lower semicontinuity and uniqueness 15
Let us continue with some general facts concerning (2.1) and rate functions.
2.3. Lower semicontinuity and uniqueness
Recall the definition of a lower semicontinuous (l.s.c.) function.
Definition 2.7. A function f : X → [−∞, ∞] is lower semicontinuous if
{f ≤ c} is closed for all c ∈ R.
Exercise 2.8. Prove that if f is lower semicontinuous then {f = −∞} is
closed.
∗ Exercise 2.9. Prove that if X is metric then f is lower semicontinuous if,
and only if, lim y→x f(y) ≥ f(x) for all x.
Say we have an arbitrary function f : X → [−∞, ∞] and would like
to produce from it a lower semicontinuous one. This is achieved by the
so-called lower semicontinuous regularization of f, which we will denote by
flsc : X → [−∞, ∞]. It is defined by
(2.2) flsc(x) = sup inf f : G ∋ x and G is open
G
.
This defines a lower semicontinuous function and in fact the maximal lower
semicontinuous minorant of f.
Lemma 2.10. flsc is lower semicontinuous and flsc(x) ≤ f(x) for all x. If
g is lower semicontinuous and satisfies g(x) ≤ f(x) for all x, then g(x) ≤
flsc(x) for all x.
Proof. flsc ≤ f is clear. To show flsc is lower semicontinuous, let x ∈
{flsc > c}. Then there is an open G containing x and such that infG f > c.
Hence by the supremum in the definition of flsc, flsc(y) ≥ infG f > c for all
y ∈ G. Thus G is an open neighborhood of x contained in {flsc > c}. So
{flsc > c} is open.
To show the last claim one just needs to show that glsc = g. For then
g(x) = sup inf g : x ∈ G and G is open
G
≤ sup inf f : x ∈ G and G is open = flsc(x).
G
We already know that glsc ≤ g. To show the other direction let c be such that
g(x) > c. Then, G = {g > c} is an open set containing x and infG g ≥ c.
Thus glsc(x) ≥ c. Now increase c to g(x).
The above can be reinterpreted in terms of epigraphs. The epigraph of
a function f is the set {(x, t) ∈ X × R : f(x) ≤ t}.