Hello, @DanielFischer. I'm still having question regarding the existence of countably additive measure $\mu$ s.t. $\mu[a,b]=F(b)-F(a)$, as $F(x)$ is defined in the problem. I can prove the case when $\mu(a,b]=F(b)-F(a)$ (proof included in my problem description), but how about the case in my problem. Sorry to bother you again, but I really would like to get this problem resolved for sake of clarity. Thank you.