Let $A, B\subset\mathbb{R}$ two Lebesgue-measurable sets of positive Lebesgue-measure. Prove that $A-B$ contains an interval. I thought that being the measure positive then the Hausdorff dimension of the two sets is 1. But I don't know how to go further. The hint says: &quot;use the convolution of the two characteristics function of the two sets&quot; but I can't really figure out how to use it.
So this homework question is in the context of $\mathbb{R}$ only, and we are using Lebesgue measure.  The sum $A+B$ is defined to be $A+B=\{a+b|a\in A,b\in B\}$. The question is: If $m(A),m(B)&gt;0$, is it necessarily the case that $A+B$ contains an open interval?  This question is quite annoying, since it is quite difficult to calculate the sum even after you construct an example. However, it seems like no matter how I constructed a set of positive measure, there is a region that have a lot of point in it (yet this region is not necessarily dense). This suggest that when summed up, the holes should be covered up somehow and thus there would be an open interval, and so the answer is yes. Sorry for a rather vague idea, but I'm just not sure what to do with this problem. I have a feeling that somehow topology would have a way to capture precisely the idea above about a region with a lot of point.  (also, I think my middle school teacher mentioned this once, and I remembered the answer to be yes; but that is quite a while ago though)  Thank you. Any help will be appreciated.  EDIT2: So Camilo's answer is gone. I am looking at the convolution method right now. Because the convolution is continuous, if it is nonzero I am done. But then there is no guarantee that the convolution is nonzero even if $x\in A+B$. I still need help with this direction. Can someone give me some hints?  EDIT: Camilo's answer below essentially suggest the method of getting 2 open interval from regularity theorem, one cover a subset of A and one cover a subset of B such that the sum of the measure of these two subset is bigger than the measure of the larger interval. user99680's answer point to a wiki which show a method that essentially simply acquire an open set. After some thinking, I believe that these method simply can't be made to work. I constructed this example in the hope that it would convince you that it is the case:  Let $I_{0},I_{1},\ldots$ be disjoint open interval such that the distance between any 2 of them is nonzero and $m(I_{n})=9^{-n}$ and all of them are inside $[0,2]$. Let $J_{0},J_{1},\ldots$ be disjoint open interval such that the distance between any 2 of them is at least $3$ and $m(J_{3n})=m(J_{3n+1})=m(J_{3n+2})=\frac{m(I_{n})}{3}$. Let $O=\bigcup I_{n}$ and $H=\bigcup J_{n}$. Construct $A\subset O$ such that $m(A\bigcap I_{n})=\frac{3m(I_{n})}{4}$. Construct $B\subset H$ such that $m(B\bigcap J_{n})=\frac{3m(J_{n})}{4}$. Now if you apply Camilo's method, it is possible that your $I$ would be one of the $I_{n}$ above, and your $J$ is one of the $J_{m}$ above. No matter which $I_{n},J_{m}$ you have, the argument does not work. Rewrite $m=3q+r$ where $0\leq r\leq 2$. If $q&lt;n$ then $m(J_{m})&gt;m(I_{n})$ so the longer interval have length $\frac{m(I_{q})}{3}$; $m(A\bigcap I_{n})$ is at most $\frac{3m(I_{q+1})}{4}=\frac{m(I_{q})}{12}$ and $m(B\bigcap J_{m})=\frac{3m(J_{m})}{4}=\frac{m(I_{q})}{4}$. Thus $m(A\bigcap I_{n})+m(B\bigcap J_{m})$ is no larger than the larger interval. If $n\geq q$ then $m(I_{n})&gt;m(J_{m})$ so the larger interval is of length $m(I_{n})$; $m(A\bigcap I_{n})=\frac{3m(I_{n})}{4}$ and $m(B\bigcap J_{m})$ is at most $\frac{3m(J_{3n})}{4}=\frac{m(I_{n})}{4}$. Once again $m(A\bigcap I_{n})+m(B\bigcap J_{m})$ is no larger than the larger interval. Hence the Camilo's argument does not work. user99680's method does not work either, since you get $O$ and $H$ which is even bigger than any interval.
How do I stop Windows 10 from updating my graphics driver? Every time Windows 10 runs updates (which I see no way to control), it updates my graphics driver.  I like the old one because it is faster.  So, I have to reinstall it.  This is getting to be a pain.  Is there any way to stop Windows 10 from updating it again? Justifying the Normal Approx to the Binomial Distribution through MGFs Would absolutely love if someone could help me with this question, in a step by step way to help those who are uninitiated to Statistics and Mathematics.   So, I am trying to "prove/justify" through MGFs how as n(the sample size) increases and goes to infinity, a standardized binomial distribution converges to the Standard Normal Distribution. So in the beginning we have $X_n $~$ Bin(n,p)$ then we standardize this $R.V.$ by subtracting the mean $E(X_n)=np$ and dividing by the $SD$ which is $np(1-p)$. After doing this, we get a new $R.V.$ with $mean=1$ and $variance=SD=1$. 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Thanks so much in advance :) Magento: Merge Product Description and Addition Information Tab How to compile .c file with OpenSSL includes? Why does my object not show up? This is meant to solve this problem once and for all. What is every reason an object isn't showing up in a render or viewport in when using the cycles render engine? Excel VBA code to turn a row with a variable number of columns into fixed-length rows I would like to change this:  Filer   ShareName       IPAddress1    IPpaddres2    IPAddress2    ... ... .... ServerA /share1/tools   192.168.1.52  192.168.1.65  192.168.1.158 ServerA /share/library  192.168.1.65  192.168.1.61  192.168.1.155 ServerB /share/tools    192.168.1.158 192.168.1.159 ServerD /share/misc     192.168.1.7 .. ... ..... .......   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This is new. It worked until now.    Works neither in Chrome nor in Firefox. i want to compile several pgfplots pictures with lualatex, since it dynamically allocates memory for compiling (in contrast to pdflatex).  In some plots, like my MWE.tex given below, i want to use a contour plot. This is typically done with gnuplot. To compile a document like this i need to use --shell-escape, such that gnuplot command can be executed while compiling the document.  \documentclass[a4paper]{article}  \usepackage[utf8]{luainputenc} \usepackage{pgfplots} \pgfplotsset{compat=newest} \begin{document} \begin{tikzpicture}   \begin{axis}     \addplot3[contour gnuplot]      {x+y};       \end{axis}   \end{tikzpicture} \end{document}   If i use  pdflatex --shell-escape MWE.tex   everything works fine, and the log says something along the lines  This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016) (preloaded format=pdflatex 2016.6.12)  13 JUN 2016 10:46 entering extended mode  \write18 enabled. .... Package pgfplots notification 'compat/show suggested version=true': document has been generated with the most recent feature set    (\pgfplotsset{compat=1.13}).  \openout3 = `MWE_contourtmp0.dat'.  \openout3 = `MWE_contourtmp0.script'. runsystem(gnuplot "MWE_contourtmp0.script")...executed.   If i use     lualatex --shell-escape MWE.tex i get the following error:  This is LuaTeX, Version 0.95.0 (TeX Live 2016)  (format=lualatex 2016.6.12)  13 JUN 2016 10:52  system commands enabled. ... Package pgfplots notification 'compat/show suggested version=true': document has  been generated with the most recent feature set (\pgfplotsset{compat=1.13}).  Package pgfplots info on input line 10: Using 'lua backend=false' for plot 0 (type 'contour external'): plot handler does not support LUA.  \openout3 = MWE_contourtmp0.dat  \openout3 = MWE_contourtmp0.script gnuplot "MWE_contourtmp0.script"  ! Package pgfplots Error: sorry, plot file{MWE_contourtmp0.table} could not be opened.   The .table file exists and looks identical to the one created by pdflatex, but it says that it cannot be executed.  (I just installed texlive 2016 yesterday, so everything should be up to date.) I'd like to get a notification when a question that I've voted to close (but isn't closed yet) is edited Email Deliverbility How can I enable permissions to edit Email Deliverability access level? Currently it is not editable and in system mode. I need to edit it and set it to "All Email so I can test Approval and workflow functionality. Using a web service is often an excellent architectural approach. And, with the advent of WCF in .Net, it's getting even better.  But, in my experience, some people seem to think that web services should always be used in the data access layer for calls to the database. I don't think that web services are the universal solution.  I am thinking of smaller intranet applications with a few dozen users. The web app and its web service are deployed to one web server, not a web farm. There isn't going to be another web app in the future that can use this particular web service. It seems to me that the cost of calling the web service unnecessarily increases the burden on the web server. There is a performance hit to inter-process calls. Maintaining and debugging the code for the web app and the web service is more complicated. So is deployment. I just don't see the advantages of using a web service here.  One could test this by creating two versions of the web app, with and without the web service, and do stress testing, but I haven't done it.  Do you have an opinion on using web services for small-scale web app's? Any other occasions when web services are not a good architectural choice? Probability that a quadratic polynomial with random coefficients has real roots So you are camping (please excuse the MMO jargon, sometimes I just can't help myself) your own particular corner of Stack Overflow, hoping for a couple of upvotes before bed.  It's freezing cold and you can't quite drag yourself across the room to grab the little blow heater, because that would mean leaving the almost adequate warmth of your laptop.  A completely objective question pops up, to which you absolutely know the answer.  Riding the surge of adrenaline, you spend the next twenty minutes typing like a madman. Desperate to beat the other lunatics that loiter in your tag of choice.  You beat them to the punch and enter a superlative answer, code samples, examples and maybe some ironic humour. You are flushed with your own awesomeness.  You go to bed. Satisfied you have garnered a couple of points of virtual self validation. You look forward to the flush of pleasure in the morning when you see your account bumped ever closer (yet never close enough) to that shining example of Skeetness.    Yet, the next morning, no such flush occurs.  Your answer sits in prime position for some questioner love and none eventuates.  You might get a couple of upvotes from some sympathetic SO compatriot, but the ultimate (I'll try to avoid any unsavory metaphors here)  satisfaction of accepted answer is denied you.    So my question. How you react to this kind of situation?   Catch yourself monitoring the questioner's account waiting for them to log on and then breaking nearby inanimate objects when your answer continues to be ignored. Panic that the questioner might delete their account, leaving your beloved question to evaporate in a puff of uninterest.  Wondering if it was something you said and answering some other of the questioner's queries in the hope of redeeming yourself in their fickle eyes.   I have seen a few examples of this type of frustration on Twitter and wondered how the hardcore Stack Overflow community dealt with this type of rejection. How to write zero in the unary numeral system In the decimal system: $...-3, -2, -1, 0, 1, 2, 3...$ In the binary system: $...-11, -10, -1, 0, 1, 10, 11...$ In the unary system: $-111, -11, -1, \text{ ??? }, 1, 11, 111...$   I was only able to find a related question but none with an answer so far: Lighter nuclei liberate energy when undergoing fusion, heavier nuclei when undergoing fission.  What is it about the nucleus of an Iron atom that makes it so stable?  Alternatively: Iron has the greatest nuclear binding energy - but why?