Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$, and $u:\mathbb{R}\to\mathbb{R}, u(x)=|x|$. Consider $\mu = \lambda \circ u^{-1}$. Find $\mu([0, a])$.
Let me start by saying there is another post with the same problem but my question is different.
Notice that $u^{-1}=1/|x|$. The answer is supposedly $\mu([0,a])=2a$. I do not understand how one arrive to this answer. This is how I will compute $\mu([0,a])$. First I have to find the set of all $u^{-1}(x)$ such that $x\in[0,a]$. But this does not make since $u^{-1}(0)=1/|0|$ which is not defined.
$$u^{-1} (A) =\{ x\in\mathbb{R} : u(x) \in A\}$$ $$u^{-1} ([0,a]) =\{ x\in\mathbb{R} : |x| \in [0,a]\}=[-a,a]$$ $$\mu ([0,a]) =\lambda (u^{-1} ([0,a]))=\lambda ([-a,a] ) =2a $$