Homogenous measure on the positive real halfline 
Define a measure $\mu\not=0$ on positive real number $\Bbb R_{>0}$ such that for any measurable set $E\subset\Bbb R_{>0}$ and $a\in \Bbb R_{>0} $, we have $\mu(aE)= \mu(E)$, where $aE=[ax;x\in E]$.


I am totally blank about this problem. I ponder on it several times but didn't get any idea. This exercise illustrates lebesgue measure's abstract and weird nature. Because if we assume E as a subset of real numbers or any interval, it totally disagrees to fulfill this translation.

I'll answer in greater generality. 
A common way to construct a measure is to take a  nonnegative  locally integrable function $w$ and define $\mu(E)=\int_E w(x)\,dx$. This does not give all measures (only those that are absolutely continuous with respect to $dx$) but for many examples that's enough.
In terms of $w$, the desired condition translates to 
$$\int_E w(x)\,dx = \int_{aE} w(x)\,dx\tag1$$ 
A way to get a handle on (1) is to bring both integrals to the same domain of integration. So, change the variable $x=ay$ in the second one, so that it becomes $\int_{E} a\,w(ay)\,dy$. Which is the same as $\int_{E} a\,w(ax)\,dx$, because the name of the integration variable does not matter. So, (1)  takes the form
$$\int_E w(x)\,dx = \int_{E} a\, w(ax)\,dx \tag2$$
or, better yet, 
$$\int_E ( w(x)-a\, w(ax))\,dx \tag3$$
So that (3) holds for every measurable set, the integrand should be zero almost everywhere. So, we need a function $w$ such that $w(ax)=w(x)/a$ for all $x$. In particular, $w(a)=w(1)/a$ for all $a$, which tells us what the function is. (The value of $w(1)$ can be chosen to be any positive number). 

The above can be generalized to produce measures such that $$\mu(aE) =a^p \mu(E)$$ for all $a>0$, where $p$ can be any fixed real number.
