The Class of Functionals which can be Represented by a Supremum

We give a characterization of all lower semicontinuous functionals on

which can be represented in the form

\mu μ

-sup

\{ f(x, u): x \in A \} { f ( x , u ) : x ∈ A }

. We also show by a counterexample that the representation above may fail if the lower semicontinuity condition is dropped.