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

Abstract:

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