Suppose that $ (A,\Sigma,m) $ is a measure space and $ H $ is a linear functional on $ {L^{\infty}}(A,\Sigma,m) $. If $$ \mathcal{U} := \left\{ u: A \to \mathbb{R} ~ \Bigg| ~ \text{$ u $ is measurable, bounded and $ \int_{A} u ~ d{m} = 1 $} \right\} $$ and there are functions $ u_{1},u_{2} \in \mathcal{U} $ such that $$ H(u_{1}) \leq 0 \quad \text{and} \quad H(u_{2}) \geq 0, $$ then my question is:
Is there a function $ u_{3} \in \mathcal{U} $ such that $ H(u_{3}) = 0 $?