In Harmonic analysis for dealing with Lorentz spaces and interpolation theorem, the Layer cake representation of $L^p$ functions plays an essential role. Let us sketch a proof here. **Theorem 1** Let $f(x)$ be a measurable function in $\Omega$ with measure $\mu$. If $f\in L^1(\Omega)$ and $f\geq 0$ almost everywhere, then \[ (1)\qquad \int_\Omega f(x) d\mu = \int_0^\infty \mu\{x\,:\, f(x)>t \}dt. \] //Proof of Theorem 1// We rewrite the right-hand side of $(1)$ as \[ \int_0^\infty \int_\Omega 1_{\{x\,:\, f(x)>t \}}d\mu dt, \] which, by Fubini's theorem, is equivalent to \[ \int_\Omega \int_0^\infty 1_{\{x\,:\, f(x)>t \}}dtd\mu = \int_\Omega \int_0^{f(x)} 1 dtd\mu = \int_\Omega f(x)d\mu. \] This completes the proof. **Remark 1** If $f\in L^p(\Omega)$, then we have \[ (2),\qquad\int_\Omega |f(x)|^p d\mu = \int_0^\infty p s^{p-1} \mu\{x\,:\, |f(x)|>s \}ds. \] To obtain $(2)$, we basically apply **Theorem 1** with $f=|f|^p$ and $t=s^p$.