Let $c>0$. Let $X \subseteq (0,\infty)$ be a Lebesgue measurable set. Define $ cX := \{ cx \mid x \in X \}. $ Then $ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$
Now I can prove this for $X$ an interval and, thus, any set generated by set operations on intervals. It is simply by using the Fundamental Theorem of Calculus and natural log $\ln$. But I'm not sure how to approach for general Lebesgue measurable set.