Integral representations for a class of operators

Let X be a completely regular Hausdorff space, E Hausdorff a quasi-complete locally convex space and C b ( X , E ) all E-valued bounded continuous functions on X with strict topologies β t , β ∞ ′ , β τ ′ . We prove that a linear continuous mapping T : C b ( X , E ) → E arises from a scalar measure μ ∈ ( C b ( X ) , β z ) ′ ( z = t , ∞ , τ ) if and only if g ( T ( f ) ) = 0 whenever g ○ f = 0 for any f ∈ C b ( X , E ) , g ∈ E ′ .