
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
April 7th, 2019, 01:52 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 $0<r<R$ we have $$\int_{E}f(x)dx\leq 2\int_{E}\left(\frac{1}{\vert B(x,r)\vert}\int_{B(x,r)}f(y)dy\right)dx.$$ Here $B(x,r)$ denotes the open ball of radius $r$ centered at $x$. I will use the following $\color{blue}{\text{Theorem}}$ from page $106$ in Stein and Shakarchi's Real Analysis: If $f$ is locally integrable in $\mathbb R^d$ then we have for almost every $x\in\mathbb R^d$ that $$\lim\limits_{\vert B\vert\to0}\frac{1}{\vert B\vert }\int_{B}f(y)dy=f(x),\quad x\in B,$$ where $B$ is a ball containig $x$. $\textbf{My Attempt:}$ Suppose for a contradiction that such $R>0$ does not exist. Then given $R>0$ there is some $0<r<R$ such that $$\infty>\int_{E}f(x)dx> 2\int_{E}\left(\frac{1}{\vert B(x,r)\vert}\int_{B(x,r)}f(y)dy\right)dx.$$ 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 $0<p<r,$ we have a contradiction of the $\color{blue}{\text{Theorem}}$. _____ Is the above proof correct? Any feedback is welcomed. Thank you for your time. 
April 7th, 2019, 04:40 PM  #2  
Senior Member Joined: Sep 2016 From: USA Posts: 600 Thanks: 366 Math Focus: Dynamical systems, analytic function theory, numerics 
The following line is wrong. Quote:
Last edited by skipjack; April 8th, 2019 at 12:20 AM.  

Tags 
functions, inequality, integrable, locally, proving, real analysis 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Integrability of the sum of integrable functions  Glo  Real Analysis  2  May 19th, 2017 06:10 AM 
locally integrable and its restriction  hubolmen  Complex Analysis  3  April 20th, 2016 10:47 AM 
integrable functions  kapital  Calculus  10  July 26th, 2012 08:36 AM 
On Continuous, Integrable Functions on [0,1]  guynamedluis  Real Analysis  8  September 30th, 2011 09:37 AM 
locally measurable functions  Mazaheri  Real Analysis  1  January 19th, 2008 06:45 AM 