Given the measure space $([0,1],\mathcal{B},\lambda|_{[0,1]})$ , I am asked to find the push forward measure $\mu$ of the measurable function $f:[0,1]\to\Bbb{R}$ given by $f(x)=x^{3}$.
My question is that how do I explicitly calculate $\mu(A)$ for any Borel set in $[0,1]$?
Now I know that $\mu(A)=\lambda(f^{-1}(A))$ . That is $\lambda(\{x\in[0,1]:x^{3}\in A\})$ . Now I don't understand how to compute the right hand side for arbitrary Borel sets $A$. But I do know how to find $\mu(a,b)$ ( or $\mu((a,b])$ ) for any interval $(a,b]$ . That is simply $\lambda(\{x\in[0,1]: a<x^{3}\leq b\}) = \lambda(\{x\in[0,1] : \sqrt[3]{a}<x\leq \sqrt[3]{b}\}) = \lambda\bigg((\sqrt[3]{a},\sqrt[3]{b}]\bigg) = \sqrt[3]{b}-\sqrt[3]{a}$ . I mention this because, I know that the collection $\{(a,b]\subset [0,1]\}$ generates the Borel Sigma Algebra on $[0,1]$ . So intuitively , I can recover $\mu(A)$ for any Borel set $A$ but how(is it possible?) to calculate explicitly what $\mu(A)$ will be?(If it is possible)
Also , I saw this post and it seems that the issue is related to that of a pdf of a random variable . So if we write it as $\displaystyle\mu(A)=\int_{A\cap[0,1]}\dfrac{1}{3x^{\frac{2}{3}}}d\lambda$ then we seem to be able to compute it. But it is still not explicitly possible to calculate this integral over all Borel sets $A$ right? .
So what is the correct description?
I also read about the Caratheodory's Extension Theorem which also allows me to extend $\mu$ from the subsets of the form $(a,b]$ to any Borel set but it still does not give an explicit description.
Edit: I am sorry if I could not explain myself properly. What I am trying to ask is: Is there a complete description of $\mu(A)$ in terms of $\lambda|_[0,1](A)$? . So if you were asked to "find the push forward measure" then what answer and in what form would you give or would consider a "valid" one?