For $k\gt 0$ and a subset $A$ of $\mathbb{R}$, let $k\,A=\{kx\mid x∈A\}$. Show that $$\mu^*(k\,A)=k \mu^*(A)$$ and that $A$ is measurable if and only if $k\,A$ is measurable.
Lebesgue measure: show that $\mu^*(k\,A)=k \mu^*(A)$
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measure-theory
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0assuming A is measurebale, I tried to use the outer measure definition.And I know that, if a set A is covered by an interval [a, b], then the corresponding set kA is covered by the interval [ka, kb].but I can't recall any path to find the answer for this. – 2012-10-21
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0Denote by $l([a,b])$ the length of the interval $[a,b]$. Is it clear that $l([ka, kb])= k\cdot l([a,b])$? – 2012-10-21