Modification of Set Function in Construction of Lebesgue Measure

Question

Suppose in the construction of Lebesgue measure we replace the set function $\mu((a,b))=b-a$ with $\mu((a,b))=\sqrt{b-a}$. What can we say about $\mu^*$ and the $\sigma$-algebra of measurable sets? Certainly $\mu^*(a,b)=\sqrt{b-a}$, and the $\sigma$-algebra consists at least of sets of Lebesgue measure $0$, but what more can be said?

Answer 1

In such case $$ \sqrt{2}=\mu\big((0,2)\big)=\mu\big((0,1)\big)+\mu\big(\{1\}\big)+\mu\big((1,2)\big)=1+\mu\big(\{1\}\big)+1. $$ Apparently, such a measure can not be positive, and, it is not hard to see, that it can not be a measure either.