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Layer cake representation
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<h2 id="applications">Applications</h2> <p>The layer cake representation can be used to rewrite the <a href="/facts/Lebesgue_integral/f4AgvC6x">Lebesgue integral as an improper Riemann integral</a>. For the measure space, ( Ω , A , μ ) {\displaystyle (\Omega ,{\mathcal {A}},\mu )} , let S ⊆ Ω {\displaystyle S\subseteq \Omega } , be a measureable subset ( S ∈ Σ ) {\displaystyle S\in \Sigma )} and f {\displaystyle f} a non-negative measureable function. By starting with the Lebesgue integral, then expanding f ( x ) {\displaystyle f(x)} , then exchanging integration order (see <a href="/facts/Fubini%2527s_theorem/q4lhtS3s">Fubini-Tonelli theorem</a>) and simplifying in terms of the <a href="/facts/Lebesgue_integral/f4AgvC6x">Lebesgue integral of an indicator function</a>, we get the <a href="/facts/Riemann_integral/vWGrPhVh">Riemann integral</a>: </p> ∫ S f ( x ) d μ ( x ) = ∫ S ∫ 0 ∞ 1 { x ∈ Ω ∣ f ( x ) > t } ( x ) d t d μ ( x ) = ∫ 0 ∞ ∫ S 1 { x ∈ Ω ∣ f ( x ) > t } ( x ) d μ ( x ) d t = ∫ 0 ∞ ∫ Ω 1 { x ∈ S ∣ f ( x ) > t } ( x ) d μ ( x ) d t = ∫ 0 ∞ μ ( { x ∈ S ∣ f ( x ) > t } ) d t . {\displaystyle {\begin{aligned}\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{aligned}}} <p>This can be be used in turn, to rewrite the integral for the <a href="/facts/Lp_space/gbad0Rv1"><i>L</i><i>p</i>-space <i>p</i>-norm</a>, for 1 ≤ p < + ∞ {\displaystyle 1\leq p<+\infty } : </p> ∫ Ω | f ( x ) | p d μ ( x ) = p ∫ 0 ∞ s p − 1 μ ( { x ∈ Ω : | f ( x ) | > s } ) d s , {\displaystyle \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,} <p>which follows immediately from the change of variables t = s p {\displaystyle t=s^{p}} in the layer cake representation of | f ( x ) | p {\displaystyle |f(x)|^{p}} . This representation can be used to prove <a href="/facts/Markov%2527s_inequality/Re85kMQJ">Markov's inequality</a> and <a href="/facts/Chebyshev%2527s_inequality/jf5ReDJv">Chebyshev's inequality</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Symmetric_decreasing_rearrangement/dIuJMW71">Symmetric decreasing rearrangement</a></li></ul> <ul><li>Gardner, Richard J. (2002). <a href="https://doi.org/10.1090%2FS0273-0979-02-00941-2">"The Brunn–Minkowski inequality"</a>. <i>Bull. Amer. Math. Soc. (N.S.)</i>. 39 (3): 355–405 (electronic). <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0273-0979-02-00941-2">10.1090/S0273-0979-02-00941-2</a>.</li> <li><a href="/facts/Elliott_H._Lieb/Vr4mrmtb">Lieb, Elliott</a>; <a href="/facts/Michael_Loss/AAwzgIaz">Loss, Michael</a> (2001). <i>Analysis</i>. <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>. Vol. 14 (2nd ed.). <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0821827833.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Willem, Michel (2013). Functional analysis : fundamentals and applications. New York. ISBN 978-1-4614-7003-8.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-4614-7003-8" target="_blank">978-1-4614-7003-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>