Abstract. We give a necessary and sufficient condition for a $J$--outer measure $\mu ^*$ on a topological space under which a set $M$ is Caratheodory $\mu ^*$--measurable iff $\mu ^*(\partial M)=0$. Under this condition we can translate the criterions of Riemann integrability in terms of $\mu ^*$--measurability. Particularly, under this condition the Lebesgue's criterion depending on the additive property of the measure is determined by the lack of the measurement precision described by the equality $\mu ^*(E)=\mu ^*(\bar E)$.

AMS Subject Classification (1991): 28C15

Received February 22, 1996 and in revised form June 14, 1996. (Registered under 6126/2009.)