1. (x) := {p ? R | U c (z) ? U c (x) + p(z ? x) for all z ? R} for x ? dom(U c ). The functions ?U (?x) and ?U c (?x) are both lowersemicontinuous on R and proper;now apply Lemma A.1 with f (x) := ?U (?x) to prove Lemmas 2.8, 2.9 and 2.10 and derive some additional properties
2. + ?x 0 for all f ? C(x 0 ) and ? > 0. For the converse direction;The fact that Assumption 3.1 implies u(x, U ) < ? for x > 0 is straightforward since we have E
3. ? U c (x 0 ) + kE[U (f )] and taking the supremum over all f ? C(x) implies u(x, U c ) ? U c (x 0 ) + ku(x, U ). So if u(x, U ) is nite for some x > 0, then u(x, U c ) is also nite. If U (0) < 0, we choose small enough such that x? > 0, x f ? C(x?) and apply the above argument to f := f + and U (x) ? U (;If U is positive on (0, ?), then x some f ? C(x) and apply (C.1) on the set {f > x 0 }. This gives E