Prove $ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$ for every Lebesgue measurable set $X$ 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.

(I dislike the title, which looks like an assignment.)
Hint:
Since you know this for intervals, use an approximation argument of step functions for the functions $x\mapsto \chi_X(x)\cdot\frac{1}{x}$ and $x\mapsto \chi_{cX}(x)\cdot\frac{1}{x}$. Where $\chi_A$ denotes the characteristic function on $A$.
