Do we have any relation between $$\mathbb{E}_X[\lvert X \rvert^2] = \int \lvert X \rvert^2 f_X(x) dx$$
and $$\mathbb{E}_{\lvert X \rvert^2}[\lvert X \rvert^2] = \int z f_Z(z) dz$$ where $Z = \lvert X \rvert^2$?
Can I apply Law_of_the_unconscious_statistician?