Show that if f is excessive, then 1 − e − f is excessive. Thus, for some purposes it is enough to look at bounded excessive functions. Show that if f and g are excessive, then f ∧ g is excessive. Let...

Show that if f is excessive, then 1 − e^{− f}
is excessive. Thus, for some purposes it is enough to look at bounded excessive functions.

Show that if f and g are excessive, then f ∧ g is excessive.

Let At be an additive functional (defined in (22.4)) and let
  Show that f is excessive.

(1) Show that every continuous function is lower semi continuous.

(2) Show that if f is lower semi continuous and x ∈ S, then

(3) Show that if f_n
is a sequence of continuous functions increasing to f, then f is lower semi continuous.

Suppose g is non-negative, bounded, and continuous, and Assumption 20.1 holds. Let g₀
= g and define
  Prove that g_n
increases to the least excessive majorant of g.

Chapter 24