The layer cake representation can be used to rewrite the
Lebesgue integral as an improper Riemann integral. For the measure space, (\Omega,\mathcal{A},\mu), let S\subseteq\Omega, be a measureable subset (S\in\mathcal{A}) and f a non-negative measureable function. By starting with the Lebesgue integral, then expanding f(x), then exchanging integration order (see
Fubini-Tonelli theorem) and simplifying in terms of the
Lebesgue integral of an indicator function, we get the
Riemann integral: : \begin{align} \int_S f(x)\,\text{d}\mu(x) &= \int_S \int_0^\infty 1_{\{x\in\Omega\mid f(x)>t\}}(x)\,\text{d}t\,\text{d}\mu(x) \\ &= \int_0^\infty\!\! \int_S 1_{\{x\in\Omega\mid f(x)>t\}}(x)\,\text{d}\mu(x)\,\text{d}t\\ &= \int_0^\infty\!\! \int_\Omega 1_{\{x\in S\mid f(x)>t\}}(x)\,\text{d}\mu(x)\,\text{d}t\\ &= \int_0^{\infty} \mu(\{x\in S \mid f(x)>t\})\,\text{d}t. \end{align} This can be used in turn, to rewrite the integral for the
Lp-space p-norm, for 1\leq p: :\int_\Omega |f(x)|^p \, \mathrm{d}\mu(x) = p\int_0^{\infty} s^{p-1}\mu(\{ x \in \Omega:|f(x)| > s \}) \mathrm{d}s, which follows immediately from the change of variables t=s^{p} in the layer cake representation of |f(x)|^p. This representation can be used to prove
Markov's inequality and
Chebyshev's inequality. == See also ==