Let $u:\mathbb{R}\to[0,\infty]$ be a Borel measurable function, define the set $S[u] = \{(x,y):0\leq y\leq u(x)\}$ and let $\lambda^n$ be the Lebesgue measure for $\mathbb{R}^n$.
I am trying to show that $\lambda^2(S[u]) = \int_{\mathbb{R}}u(x)\lambda(dx)$. From Tonelli's Theorem we have $$ \begin{align*}\lambda^2(S[u]) &= \int_{\mathbb{R}^2} \chi_{\{0\leq y \leq u(x)\}}(x,y)\lambda^2(dx,dy)\\ &= \int_\mathbb{R}\int_\mathbb{R} \chi_{\{0\leq y \leq u(x)\}}(x,y) \lambda(dy)\lambda(dx)\\ &= \int_\mathbb{R}\Big(\int_0^{u(x)} \lambda(dy)\Big)\lambda(dx)\\ &= \int_{\mathbb{R}}u(x)\lambda(dx)\end{align*}$$ Looking for proof verification only.