An example concerning the Yosida-Hewitt decomposition of finitely additive measures

Abstract

Let λ \lambda be Lebesgue measure on the Lebesgue σ \sigma -algebra L \mathcal {L} of I := ] 0 , 1 [ I:=]0,1[ . The author gives an example of a purely finitely additive measure φ : L → [ 0 , 1 ] \varphi :\mathcal {L} \to [0,1] vanishing on λ \lambda -null sets such that ∫ f d φ = ∫ f d λ \smallint f\,d\varphi = \smallint f\,d\lambda for every bounded continuous function f on I ( f ∈ C b ( I ) ) (f \in {C_b}(I)) . Consequently, λ − φ ∈ L ∞ ( λ ) ′ \lambda - \varphi \in {L^\infty }(\lambda )’ annihilates C b ( I ) {C_b}(I) and is not purely finitely additive, contrary to an assertion of Yosida and Hewitt.