It is known that:
If $\cal{A} \subset \mathbb{R}^n$ is Lebesgue measurable, and $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear mapping, then $L(\cal{A})$ is Lebesgue measurable and \begin{align} \mu(L({\cal{A}})) = |\det L|\mu(\cal{A}), \end{align} where $\mu({\cal A})$ is the Lebesgue measure of ${\cal A}$, and $|x|$ returns the absolute value of a real number $x$.
But what if the mapping is nonlinear? In particular, if $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a continuously differentiable nonlinear mapping (but is not bijective), what the value $\mu(F({\cal{A}}))$ should be? Are there any textbooks for explaining it? Thank you very much!
In the case where $F$ is injective, you have $\mu(F(A)) = \int_A |\det DF| \; d\mu$ where $DF$ is the Jacobian matrix of $F$. If it's not injective, find (if possible) a measurable set $B$ such that $F$ is one-to-one on $B$ and $F(B) = F(A)$, and then $\mu(F(A)) = \mu(F(B)) = \int_B |\det DF|\; d\mu$.