Layer cake representation

Applications

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 L**p-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.

References

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References

1. (2013). "Functional analysis : fundamentals and applications".