Let $\mu_i$ be the Lebesgue measure . Let $d\mu_0 = dx$ ; $0\leq a\leq b \leq 1$ and consider :
$\mu_1 [a,b]= \int_{a}^{b} x d\mu_0$
$\mu_2[a,b]= \int_{a}^{b} x d\mu_1$
And so on until :
$\mu_k[a,b] = \int_{a}^{b} x d\mu_{k-1}$
Then what is $\mu_k$?
I haven’t studied any theorems involving derivatives like Radon Nikodym yet , but my professor said that they are not necessary for this.
My thought : I can prove the Fundamental Theorem of Calculus for Lebesgue integrals then evaluate $\mu_1$ and the rest follows by induction .
Is this the right way ? If it is not , any hints on how to proceed ?
A first step is to express $\int f(x)d\mu_k$ for a bounded measurable function $f\colon [0,1]\to \mathbb R$. One can show (by approximating by simple functions), that $$ \int f(x)d\mu_k=\int x f(x)d\mu_{k-1}. $$ One can treat the last integral by applying the previous reasoning with $k$ replaced by $k-1$ and $x\mapsto f(x)$ by $x\mapsto xf(x)$ to get $$ \int f(x)d\mu_k=\int x^2 f(x)d\mu_{k-2}. $$ Then the pattern appears.