Let $\mu$ be an absolutely continuous measure with respect to the Lebesgue measure on $\mathbb{R}$ , and $f:\mathbb{R}\to \mathbb{R^+}$ its Radon-Nikodym derivative . We can write $\int_a^bf(x)dx$ in term of $\mu$ by the formula $$\int_a^bf(x)dx=\mu\left[a,b \right].$$ Now for some $p\geq 0$, can we write $\int_a^b(f(x))^pdx$ in term of the measure $\mu$ only?
I tried $$\int_a^b(f(x))^pdx=\int_a^b(f(x))^{p-1}f(x)dx=\int_a^b(f(x))^{p-1}d\mu(x).$$ But I still have the term $f$ in the formula, I want to write this only in term of $\mu$. I don't know if it is possible.