Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
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

2 Answers 2

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.

share|improve this answer

(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 $A$.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.