# measure theory -Lebesgue measure problem 1

For $k>0$ and $A$ is a subset of $\mathbb R$,let $kA=\{kx:x∈A\}$

Show that $m^{*}(kA)=k m^{*}(A)$ $A$ is measurable if and only if $kA$ is measurable.

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Have you seen the definition of a measurable function yet? –  icurays1 Nov 19 '12 at 6:30
It is true for intervals. Use the definition of outer measure. –  leo Nov 20 '12 at 21:32

Consider the function $f(x)=x/k$. f is measurale therefore $kA=f^{-1}(A)$ is also measurable.
Now repeat the same argument using the inverse of $k$