Some topics in complex and harmonic analysis

for all f ∈ Cb(R ) and λ is continuous in the sense that if {fj} ∞ j=1 is a sequence of bounded continuous functions on R which is uniformly bounded and converges uniformly on compact subsets of R to a bounded continuous function f , then {λ(fj)} ∞ j=1 converges to λ(f). If f is any bounded continuous function on R, then there is a sequence {fj} ∞ j=1 of continuous functions on R n such that each fj has compact support in R , the fj ’s are uniformly bounded, and the fj ’s converge to f uniformly on compact subsets of R . As a result, a finite measure on R is determined by its restriction to the vector space of continuous functions with compact support on R. In fact, if one starts with a linear functional on the vector space of continuous functions