Integrals which are convex functionals

LEMMA 1. Suppose f(t, x) = F(x) for all t9 where F is a lower semi-continuous proper convex function on R n. Then f is a normal convex integrand.

LEMMA 2. Suppose f is a convex integrand such that f(t, x) is measurable in t for each fixed x, and such that, for each t, f(t, x) is lower semi-continuous in x and has interior points in its effective domain {x\f(t, x) < +°°}. Then f is a normal convex integrand.

LEMMA 3. Let fbe a normal convex integrand with conjugate /*. Then, for every measurable function u* from T to R n, the function /*(ί, u*(t)) is measurable in t.

LEMMA 4. Let f be a normal convex integrand. Let z be a measurable function from T to R n. Then the functions prox (z(t) \ ft) and prox (z(t) \f*) are measurable in t.

Lemma 3 and the normality of g. The limit can be taken over a count- able sequence in λ, so <α, V#?(0)> is measurable in t. It follows that prox (z(t) I /ί) is measurable in £, and likewise prox (z(t) \ft*) because prox (z(t) \f*) = z(t) - prox (z(t)\ft)

LEMMA 5. If f is a normal convex integrand, then f* is a normal convex integrand, too.

LEMMA 6. Let f be a normal convex integrand. Let a be a measurable real-valued function on T such that inίx f(t, x) < a(t) for every t . Then there exists a measurable function u from T to R n such that f{t, u{t)) ^ a(t) for every t .

THEOREM 1. Let fbe a normal convex integrand. Suppose there exists at least one u* e L* such that f*(t, u*(t)) is a summable func- tion of t. Then = \ f(t, u{t))dt, ueL , is a well-defined convex function on L with values in (-co, +<*>].

THEOREM 2. Suppose L and L* are decomposable. Let f be a

Lemma 5, and they are proper by the hypothesis. For any x* in R n we have f(t, x) + /*(ί, x*) ^ <x, £*> by conjugacy. Hence, for any ueL and u*eL*, If(u) + I^(u*) = \ fit, u(t))dt + ( /*(ί, u*(t))dt JT JT ^ ( <u{t), u*(t)>dt = <u, u*> . JT It follows that If*(u*) ^ sup {(u, u*y — If(u) I u e L} = -inf {If(u) - <u, u*}\ueL}

THEOREM 3. Let T be of finite measure. Suppose that L* is decomposable, and that L is actually Lζ(T). Let fbe a normal convex integrand satisfying the following condition: there exists some ae L and ε > 0 such that, for each xeR n with \ x | < ε, the function f(t, a(t) + x) is finite and bounded in t. Then If on L and If* on L* are convex functions conjugate to each other. Moreover, If is con- tinuous at a in the norm topology of L — LZ(T).

THEOREM 4. Let T be of finite measure. Let f(t, x) be a finite convex function of x for each t and a bounded measurable function of t for each x. Then If is a well-defined finite convex function on L™(T) which is everywhere continuous with respect to the uniform norm. Moreover, the conjugate (If)* of If on Lζ(T)*, the space of all linear function as on L~(T) continuous with respect to the uniform norm, is given by //sK in the following sense: if v e Lζ(T)* is of the form

COROLLARY 1. Under the hypothesis of Theorem 4, the convex set {u* e Li(T) I (If*)(u*) + <a, u*> + a £ 0} is weakly compact (with respect to the pairing between Un{T) and Ln(T)) for any aeL~(T) and any real number a.

COROLLARY 2. Let D be a subspace of Lζ(T) supplied with a locally convex topology at least as strong os the uniform norm topology, and let D* be the space of continuous linear functionals on D. Suppose that no nonzero linear functional on Lζ(T) of the form \ <u(t),u*(t)ydt , u*eL x n(T) , r vanishes throughout D. Then, under the hypothesis of Theorem 4, // is a continuous finite convex function on D, and the conjugate (//)* of If on D* is given by If*, in the sense that if v e D* corresponds to

Corollary 2 is applicable, of course, to various situations where T has topological or differentiate structure, and D is a space