Uniform separation of points and measures and representation by sums of algebras

Abstract

LetX andY _i, 1 ≦i ≦k, be compact metric spaces, and letρ _i:X →Y _i be continuous functions. The familyF={ρ _i} ^1/k _i is said to be ameasure separating family if there exists someλ > 0 such that for every measureμ inC(X)*, ‖μ ^o ρ ⁻¹ _i ‖ ≧λ ‖μ ‖ holds for some 1 ≦i ≦k.F is auniformly (point) separating family if the above holds for the purely atomic measures inC(X)*. It is known that fork ≦ 2 the two concepts are equivalent. In this note we present examples which show that fork ≧ 3 measure separation is a stronger property than uniform separation of points, and characterize those uniformly separating families which separate measures. These properties and problems are closely related to the following ones: letA ₁,A ₂, ...,A _k be closed subalgebras ofC(X); when isA ₁ +A ₂ + ... +A _k equal to or dense inC(X)?