Ture or False? If $\sigma(x)=x^2$ on an interval $[a,b]$ and if $\sigma^*$ is the Borel measure on $[a,b]$ that is induced by $\sigma$, then a Borel set $E$ in $[a,b]$ is a null set for $\sigma^*$ if and only if $E$ is a Lebesgue null set.
Well, I don't quite get what $\sigma^*$ really does. If $A\subseteq [a,b]$ then how does one define $\sigma^*(A)$? Would it be $b^2-a^2$? I guess I need some explanation on the word "induced". Thanks in advance for any help.
Definition of $\sigma^{*}$: $\sigma^{*}(A)=m \{x: x^{2} \in A\}$ where $m$ is Lebesgue measure. The fact that $ m(A)=0$ iff $\sigma^{*} (A)=0$ is a consequence of the following theorem: if $f$ is absolutely continuous then $f$ maps sets of measure $0$ to sets of measure $0$. The function $f(x)=x^{2}$ is absolutely continuous because it is the indefinite integral of $x \to 2x$. Simialrly, use the fact that the function $\sqrt {|x|}$ is integrable for the converse part.