I've been struggling with this exercise, and was hoping someone here might be able to point me in the right direction.  Question. Let $m^*$ denote the Lebesgue outer measure on $[0,1]$. Show that there exists $A \subseteq [0,1]$ s.t. for every interval $I \subseteq [0,1]$ we have   $m^*(A \cap I)&lt;m^*(I)$, $m^*(A^\complement \cap I)&lt;m^*(I).$     EDIT: I guess I misunderstood the conditions, and this is indeed a duplicate. Apologies.
I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere.   To be precise, if $\mu$ denotes Lebesgue measure, how would one construct a Borel set $A \subset \mathbb{R}$ such that $$0 &lt; \mu(A \cap I) &lt; \mu(I)$$ for every interval $I$ in $\mathbb{R}$?   Moreover, would such a set necessarily have to contain infinite measure?
There seems to be more than a few computer science/programming Stack Exchange networks (is that the correct term?). Stack Overflow, being the first, has by far the most users, questions, and answers.  What is the reasoning for creating the others, and are there clear guidelines for which kinds of questions should be posted? I can see a large amount of potential overflow, many cases of people not getting a good answer to their question, because the person who has the answer isn't browsing that particular network at the moment.  I understand that they were probably created for organizational purposes, but wouldn't it almost make more sense to just have them as categories under Stack Overflow, keep them separated but still connected, instead of making people have to create multiple 'account's, one for each network?  I am sure there was a very good reason to break them up, but as someone that is new to SE, it can be somewhat intimidating to decide which one to post in to ensure you get a good answer. For example, if I am a computer science student, my first instinct might to be to post in the computer science network, until I see that it literally has 1% of the users as the Stack Overflow network, which still seems to be for programming/computer science related questions.  So my second instinct would then to be to post it into the Stack&nbsp;Overflow site on the basis that I have a significantly higher chance of my question being seen and getting a good answer. Then I notice that there is also a 'programmers' network, and I don't even begin to know where that fits in. I, and I am assuming most people, will probably just post on Stack&nbsp;Overflow to be safe.  My question is, other than the short little description blurb of each one, is there a clear cut set of guidelines which what each network is intended for, what kinds of questions should go to each one, and is anything being done to encourage people to post in these newer, smaller (more specialized?) networks as opposed to just posting in big daddy Stack Overflow?  Now again, this is for computer science/programming/"why isn't this code doing what I want?" related questions, I am not saying that if I had a question about Linux or WordPress or something I would have the same confusion. Why must $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ -- the set of all polynomials in $\sqrt{2}$ and $\sqrt{3}$ with rational coefficients  -- contain multiplicative inverses?   I have gathered that every element of $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$ takes form $a + b\sqrt{2} + c\sqrt{3} + d\sqrt{2}\sqrt{3}$ for some $a,b,c,d \in \mathbb{Q}$, and I have shown that $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{3}]$ are both fields, but it's not clear to me why this allows for general multiplicative inverses to exist in $\mathbb{Q}[\sqrt{2}, \sqrt{3}]$. Reference undefined in \section with amsbook I have a section that is devoted to the proof of a theorem, and hence I used  \section{Proof of Theorem~\ref{thm}}   Of course, because of the headers that are capitalized, it asks for \ref{THM} when producing the header, and I get an obvious error. What is the proper way to handle such a case? Below is a MWE:  \documentclass{amsbook} \usepackage{lipsum} \newtheorem{thm}{Theorem} \begin{document} \chapter{First Chapter} \section{The Theorem} \begin{thm}[Lipsum Theorem] \label{thm} \lipsum[1] \end{thm} \section{Proof of Theorem~\ref{thm}} \lipsum[1-23] \end{document}   When compiling I get:  LaTeX Warning: Reference `THM' on page 3 undefined on input line 14.   (each time it produces a header), and the header shows     `2. PROOF OF THEOREM ??' Why bar magnet can produce magnetic field? When we canonically quantize the scalar field in QFT, we use a Lorentz invariant integration measure given by $$\widetilde{dk} \equiv \frac{d^3k}{(2\pi)^3 2\omega(\textbf{k})}.$$  How can I show that it is Lorentz invariant? Is it possible to cancel or clear a notification created by using notify-send? So I wrote a little script that makes sure a certain user plugs in a laptop when he logs on (it disables if he doesn't). The script uses notify-send to tell him to plug it in. If he plugs it in, the script quits. Is it possible to clear the notification automatically when he plugs it in? I'm thinking it might require somehow getting the process id spawned by notify-send and killing that PID, but I don't know how to do this.  Here's the current script:  #!/bin/bash   cat /sys/class/power_supply/BAT0/status OUTPUT="$(cat /sys/class/power_supply/BAT0/status)" echo "${OUTPUT}" if [ "${OUTPUT}" = "Charging" ] || [ "${OUTPUT}" = "Unknown" ]; then     echo charging or full elif [ "${OUTPUT}" = "Discharging" ]; then     notify-send -i /home/evamvid/Documents/Programming/OokiNoUse/power25.png "Hey there brother" "plug it in"     COUNTER=0 while [ "$COUNTER" -le 12 ] do     cat /sys/class/power_supply/BAT0/status     OUTPUT="$(cat /sys/class/power_supply/BAT0/status)"     echo "${OUTPUT}"     if [ "${OUTPUT}" = "Charging" ] || [ "${OUTPUT}" = "Unknown" ]; then         exit     elif [ "${OUTPUT}" = "Discharging" ]; then         COUNTER=$(($COUNTER+1))         echo $COUNTER         sleep 1     fi done fi How to change geometry for one page without using \newgeometry or \restoregeometry "An error occurred when reviewing this item" On approving an edit, I just get this pop up since morning. At first I thought there might be some internet connection issue, but this is now the 3rd to 4th time I got the same error; also, the review is accepted because on refresh I can see my name with approved text beside, but it still throws this error.    I am on a Windows 7 machine, running Firefox 24 (Icognito/Private Mode)    Also, this happens only when edit(1) is written on an question/answer, I don't go to review queue and approve, these reviews were directly loaded from the notification beneath the question/answer.  Martijn Pieters reproduced the problem from the review queue three times; each time the improve option was picked, accepting the suggested edit but editing the post some more. The network console shows that posting to https://stackoverflow.com/review/task-reviewed fails, the server responds with a 404 error:  Request headers: POST /review/task-reviewed HTTP/1.1 Host: stackoverflow.com Connection: keep-alive Content-Length: 137 Accept: application/json, text/javascript, */*; q=0.01 Origin: http://stackoverflow.com X-Requested-With: XMLHttpRequest User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10_7_5) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/31.0.1650.63 Safari/537.36 Content-Type: application/x-www-form-urlencoded DNT: 1 Referer: https://stackoverflow.com/review/suggested-edits/3575396 Accept-Encoding: gzip,deflate,sdch Accept-Language: en,en-US;q=0.8,nl;q=0.6,nb;q=0.4,no;q=0.2,de;q=0.2,fr;q=0.2 Cookie: &lt;redacted&gt;  POST data: taskTypeId=1&amp;reviewTaskId=3575396&amp;taskResultTypeId=5&amp;fkey=f8791a329b445b658876f4d3afe7961e&amp;postCommentId=null&amp;taskResultSubtypeFlags=null  Response headers: HTTP/1.1 404 Not Found Cache-Control: private Pragma: no-cache Content-Type: application/json X-Frame-Options: SAMEORIGIN Date: Fri, 13 Dec 2013 12:04:53 GMT Content-Length: 0   On the  itself, no record is made of the accept; instead the suggested edit is recorded as approved by 3 people not including Martijn Pieters. In this case the improve review arrived too late.  In , improve again resulted in the error, but the Community account is shown rejecting the edit. My Mac Book Pro 13.-" doesn´t get past the grey booting screen with the apple and the spinning wheel (The loading bar appears but when it´s done nothing happens)   I have 220 GB in one single Partition (Macintosh HD) and there is all my work my music and the only copy of ALL MY PHOTOS. I´m so stupid I don´t have a backup of anything but know I have an external HD with 300GB space. I would like to repair my HDD but more important I need to save my files but I don´t know how to do it in Disk Utility Mode. I read there is a Diskwarrior but I don´t know how to install it without booting the mac. Please help!!!  When I start on Single User mode and enter /sbin/fsck -fs this appears  Executing fsck_hfs (version diskdev_cmds-557-393)  Checking Journaled HFS Plus volume.  The volume name is Macintosh HD  Checking extents overflow file.  Checking catalog file  Incorrect number of thread records  (4, 23745)  Checking multi-linked files.  Checking catalog hierarchy  Invalid volume directory count (It should be 175703 instead of 175701) Checking extended attributes file.  Incorrect number of extended attributes (It should be 340062 instead of 340064)  Checking volume bitmap.  Checking volume information.  The volume Macintosh HD could not be repaired after 3 attempts.  *****The volume was modified *****   What are the next steps if fsck fails to repair a drive? Apache2 restart failed in Ubuntu 16.04 newenvironment: error with \begin{align} and \end{align} Is it appropriate to buy a "thank you" gift for a PhD supervisor? I can show that for $x &gt; 0$ and $r_{i} &gt; 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ However, I can't do this using straight up induction, strong or weak. Can someone do this?   You can show the inequality holds for $n$ if it holds for $n+1$ using $r_{n+1} = (r_1 \cdots r_n)^{1/n}$, so going down isn't a problem, but going up has eluded me. Additionally, you can show that if it holds for $n=a$ and $n=b$ then it holds for $n=ab$. Using powers of two and the above is sort of a way to prove the above by induction but it certainly isn't "normal".  You can also prove the above using Lagrange multipliers (which isn't surprising). The last proof I know is where you compare coefficients of $x^k$ and use the AM-GM inequality. The difference between log and ln $$\dfrac{1}{2}\ln(x+7)-(2 \ln x+3 \ln y)$$  Our professor let's us solve this but i do not understand how $\ln$ works. He says it has same properties with $\log$ but i still don't get it. What's the difference of the two? Create a table with LaTeX What is the command to remove all files but not directories? Let's say I have a directory tree like this:  FOLDER:     file1     file2     file3     Subfolder1:         file1         file2     Subfolder2:         file1         file2   If I used rm -r FOLDER/*, everything in FOLDER would be deleted including sub-directories. How can I delete all files in FOLDER and in its sub-directories without deleting actual directories? Prove that $2730$ divides $n^{13} - n$ for all integers $n$. Prove that $2730$ divides $n^{13} - n$ for all integers $n$.   What I attempted is breaking $2730$ into $2, 3, 5$, and $7, 13$. Thus if I prove each prime factor divides by $n^{13} - n$ for all integers $n$, it will prove the original problem.   I already proved $2, 3$, and $5$ work. Struggling with how to do $91$?  Other solutions are welcome, but they should be associated with mod and solutions of this kind, because my answer should pertain to what was learned in class.  (Edit: Fixed the 91 problem, being slow today...In any case, I've got this problem now, so no answers are necessary. Posting my own soon.) Control custom properties How can I add remove and create a callback/event for a custom property using bpy. I have a scene, called scene, and I want to add a custom property named x, defulat value 2 (as intereger), and a callback foo, to run when it is changed. I also need to be able to read the value from it, and delete it. How do I do this? How did Harry become a Horcrux? Cannot get 2 raster files of same cell size to align ArcMap 10.2