I need to determine the pushforward measure $f_*m$ of continuous measurable functions like e.g $f(x)=c,f(x)=x,...,f(x)=cos(x)$ and also if possible i need to calculate the radon Nikodym derivative with respect to $m$, where $m$ denotes the Lebesgue measure. I would probably not struggle as much as i do, if i had an example of how its done, since i cant really find any easy examples on the internet such as : $f(x)=x^2$.
Any sort of advice will be very helpful and thanks in advance.
Working the details requires work, but here is an idea.
We have, where $f$ is monotone, $$m(f^{-1}(a,b))=f^{-1}(b)-f^{-1}(a)=(f^{-1})'(\xi)\,(b-a)=(f^{-1})'(\xi)\,m((a,b)).$$
For a continuous positive function $g$ and appropriate partitions $\{a_k\}$, $$ \int g\,d(f_*m)=\lim_n\sum_k g(a_k)\, m(f^{-1}(a_k,a_{k+1}))=\lim_n\sum_kg(a_k)\,\frac1{f'(f^{-1}(a_k))}\,m((a_k,a_{k+1}))=\int \,g\,\frac1{f'\circ f^{-1}}\,dm. $$