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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.

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Use substitution aka transformation formula for integrals. – martini Oct 5 '12 at 13:14
Have you done the analogous proof $\int_X dx = \int_{X+c}dx$ – GEdgar Oct 5 '12 at 14:07
@martini Yes, I think that's a very clever way. But unfortunately I know no thoerem of that kind in the context of abstract measure theory. Do you know any? And maybe a reference book? – julypraise Oct 5 '12 at 14:28
@GEdgar Yes, though not exactly your eqn but kinda simialr, namely, $\int_{X} f(x) dx = \int_{X+c} f(x-c) dx$. I know this holds by the translation invariance property of Lebesgue measure. – julypraise Oct 5 '12 at 14:29
@julypraise Please use proper English. – AD. Oct 6 '12 at 6:34

Suppose $m$ and $n$ are non-negative measures and $c$ is a positivie number and $n=m/c$. Can you show that $$ \int_A f\,dm = \int_A (c f)\,dn\text{ ?} $$ If you can, let $A=cX$, $m=$ Lebesgue measure, $f(t)=1/t$. Then find a one-to-one correspondence between $A=cX$ and $X$ such that the value of $1/t$ for $t\in X$ is the same as the value of $cf(t)$ for $t\in A$, and think about that.

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(I dislike the title, which looks like an assignment.)


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 $A$.

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Don worry, this is not an assignment question. I'm kind of new to this subject. And I think I haven't learned that techniuqe of using step functions yet. Would you give me acutally full description of this technique, or a theorem usable realted to this technique, or reference text? Thanks! – julypraise Oct 5 '12 at 14:35
@julypraise What I meant was that you should try to avoid phrases like "Prove this...", "Do this..." – AD. Oct 5 '12 at 21:33
@julypraise What book do you read? it is good to know some dense spaces, and it is also very common to have such results in the text books. – AD. Oct 5 '12 at 21:36

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