Denote by $m^n$ the $n$-dimensional Lebesgue measure. Let $A\subset\mathbb{R}^n, x\in\mathbb{R}^{n-1}$. Define $P_n(x) :=\{x + e_nt: t\in\mathbb{R}\}$ for the standard basis vector $e_n$. Then $P_n$ gives a line perpendicular to the hyperplane $\mathbb{R}^{n-1}$. Suppose that $A\subset\mathbb{R}^n$ such that $m^n(A)<\infty$. I would like to conclude rigorously that
$$m^n(A)=\int_{\mathbb{R}^{n-1}}m^1(A\cap P_n(x))dx$$
but I am not sure how. The claim feels really intuitive because what we are effectively doing is that we are moving a "laser beam" across the $(n-1)$-dimensional hyperplane $\mathbb{R}^{n-1}$ and summing how much beam fits inside the set $A$.