 My Math Forum Proving an Inequality for Locally Integrable Functions

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 April 7th, 2019, 02:52 PM #1 Newbie   Joined: Apr 2019 From: Madison, Wisconsin Posts: 2 Thanks: 0 Math Focus: Measure Theory & Integration Proving an Inequality for Locally Integrable Functions >$\textbf{The Problem:}$ Let $f\geq 0$ be a bounded function and $E\subset\mathbb R^d$ have finite measure. Prove that there exists $R>0$ such that for all $00$ does not exist. Then given $R>0$ there is some $0\int_{E}f(x)dx> 2\int_{E}\left(\frac{1}{\vert B(x,r)\vert}\int_{B(x,r)}f(y)dy\right)dx.$$In particular, this means that for almost every x\in E we must have$$\infty>\frac{f(x)}{2}>\frac{1}{\vert B(x,r)\vert}\int_{B(x,r)}f(y)dy.$$And since by the assumption that f\geq0 and bounded, we have that$$\frac{1}{\vert B(x,r)\vert}\int_{B(x,r)}f(y)dy\geq \frac{1}{\vert B(x,p)\vert}\int_{B(x,p)}f(y)dy$$for all 0 April 7th, 2019, 05:40 PM #2 Senior Member Joined: Sep 2016 From: USA Posts: 685 Thanks: 461 Math Focus: Dynamical systems, analytic function theory, numerics The following line is wrong. Quote:  Originally Posted by Gaby Alfonso Then given R>0 there is some 0\int_{E}f(x)dx> 2\int_{E}\left(\frac{1}{\vert B(x,r)\vert}\int_{B(x,r)}f(y)dy\right)dx.$$ In particular, this means that for almost every$x\in E\$ we must have $$\infty>\frac{f(x)}{2}>\frac{1}{\vert B(x,r)\vert}\int_{B(x,r)}f(y)dy.$$
This is not true. You have taken an inequality for integrals and tried to use it to imply a pointwise inequality which in general is not valid even on a set of full measure.

Last edited by skipjack; April 8th, 2019 at 01:20 AM. Tags functions, inequality, integrable, locally, proving, real analysis Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Glo Real Analysis 2 May 19th, 2017 07:10 AM hubolmen Complex Analysis 3 April 20th, 2016 11:47 AM kapital Calculus 10 July 26th, 2012 09:36 AM guynamedluis Real Analysis 8 September 30th, 2011 10:37 AM Mazaheri Real Analysis 1 January 19th, 2008 07:45 AM

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