I have been struggling with the following for quite some time now. If anyone can give me some help, it will be much appreciated:  Let $f$ bounded, measurable and $E$ be a set of finite measure. Let $A \subset E$ be measurable. Prove that:  $\displaystyle \int_{A} f = \int_{E} f \chi_A $
Let $f$ be a bounded measurable function on a set of finite measure $E$.  For a measurable subset $A$ of $E$, show that $\int_A f = \int_E f \cdot \chi_A$     Lemma: Let $f$ be a bounded measurable function on a set of finite measure $E$.  Suppose $A$ and $B$ are disjoint measurable subsets of $E$.  Then $\int_{A \cup B} f = \int_A f + \int_B f$.   My first approach was to view the problem like this utilizing the above lemma:  $$\int_E  f \cdot \chi_A = \int_{E\setminus A} f \cdot X_A + \int_A f \cdot \chi_A $$  I'm not really sure how the result follows from the above.
I am trying to mount my ext4 (sdc6) partition from a live usb because that's how I deleted two unused partitions left from a prior install of ubuntu, using GParted. The swap is mounted. However, I can't edit fstab, or find anywhere to click in nautilus to mount sdc6. In the terminal, I get the error: "can't find /dev/sdc6 in fstab". I'm afraid to exit the usb and update grub if that partition isn't mounted. Is there any easy way to do this, or do I have to install gksu or some other editor? Or do I need to mount it as sda6?  Device     Boot Start     End Sectors  Size Id Type /dev/sda1  *     2048 3905535 3903488  1.9G  b W95 FAT32  Device     Boot     Start        End   Sectors   Size Id Type /dev/sdc1            2048   29362175  29360128    14G 27 Hidden NTFS WinRE /dev/sdc2  *     29362176   29566975    204800   100M  7 HPFS/NTFS/exFAT /dev/sdc3        29566976  487023432 457456457 218.1G  7 HPFS/NTFS/exFAT /dev/sdc4       487024638 1250263039 763238402   364G  5 Extended /dev/sdc5       728393728  740290559  11896832   5.7G 82 Linux swap / Solaris /dev/sdc6       487024640  728391679 241367040 115.1G 83 Linux  Partition table entries are not in disk order. I was so excited about the Spring '14  as a way to present data visualizations on a public site, and now it looks like that's not possible.  I created a page that looks basically like the release notes sample:  &lt;apex:page &gt;     &lt;analytics:reportChart reportId="00OW0000000Spru"&gt;&lt;/analytics:reportChart&gt; &lt;/apex:page&gt;   I tested it inside the authenticated environment, where it works fine, then added it to a Site. There, I get the error: To view the report chart, you need both the Run Reports permission and access to the report's folder. But, Run Reports is not a permission I can grant to the Site Guest User.  Anybody have any ideas for a workaround? Or is it not going to be possible to show report charts on a public site? Expansion of $(1+\sqrt{2})^n$ I was asked to show that $\forall n\in \mathbb N$ there exist a $p\in \mathbb N^\ast$ such that $$(1+\sqrt{2})^n = \sqrt{p} + \sqrt{p-1}$$   I used induction but it wasn't fruitful, so I tried to use the binomial expansion of $(1+\sqrt{2})^n$ but it seems I lack some insight to go further.   Any hint is welcomed. Rewrite Sharepoint URL (Sharepoint 2013) I'd like to rewrite all my site urls in the way the the user can't figure out the exact name of the the page his viewing. let's say that my site is "www.justarandomsite.com"  and may page is "page.aspx" I' like to have instead of "www.justarandomsite.com/pages/page.aspx" sothing like this: "www.justarandomsite.com/skdhfjksdgfjksdgfjdsgvdjhsgfjdsdsjfgkdsjgfkdsgfjdsgfjkdsg" For symmetric stable distributions, why is $\alpha \le 2$? What are PATH and other environment variables, and how can I set or use them? Questions about    setting environment variables the PATH   are very common here, and in most cases the answers are very similar to each other. In the future it would be nice to have a good Q/A for this.  So the question is: What are environment variables, like the executable PATH, and how can I change and use them on major operating systems?  A good answer would include a simple explanation of what environment variables and especially PATH mean to the OS, as well as simple guidelines on how to set and read them accordingly. I have a failure to transfer my game from one HDD to another, and one PC to another. I copied the entire Skyrim folder from one HDD to another. And told where to find it also. Steam did a huge check of my HDD.  But it ended up downloading most of the files anyway. It took a week to download Dawnguard, Hearthfire, and Dragonborn files along with my Hi-Res textures.  I did ask about it in chat. But I feel this could help me or somebody else.  This is the screen I am looking at after transferring my entire hard drive to to new one. Steam downloaded my games, including Skyrim even though my Skyrim data folder had all the data files for Dawnguard, Hearthfire, Hi-Res and Dragonborn.  I have my copied files and have got rid of everything not steam related. I just have my DLCs and they include my hires files.  This is what it looks like.    But I still get this :     What could be causing this? How can I rectify? And in future when one is migrating to a new PC, how can this be avoided?  EDIT::::::::::::::::: Removed everything : Here is how it looks :  Still the prompt as before Steam Error 15  (Side Note : I transferred all my Origin games just by copy/pasting. This really is not nice.)    EDIT 2 :::::::::::::::::::::  And one more EDIT  I tried getting game cache up to speed, but this is prompt I am getting. But I played the game, it was working, I could pick out my character and all, and escape from Helgen.  EDITED 3 :::::::::::::::::  How does one transfer a Steam game? A game that is heavily modded, will the backup a Steam Game option suffice? I'm writing a simple bash script, but I need it to check whether it's being run as root or not. I know there's probably a very simple way to do that, but I have no idea how.  Just to be clear: What's a simple way to write a script foo.sh, so that the command ./foo.sh outputs 0, and the command sudo ./foo.sh outputs 1? Drupal unexpected "Fatal error: Class name must be a valid object or a string [...] common.inc" on every page "Fatal error: Class name must be a valid object or a string in C:\xampp\htdocs\uc\includes\common.inc on line 7506"  The error above showed up unexpectedly. I was adding a "Long text" field to a content-type, but when I expected to be brought to a saved field configuration page, I instead got a Fatal error on every page.  I had a look at includes\common.inc:  /**  * Get the entity controller class for an entity type.  */ function entity_get_controller($entity_type) {   $controllers = &amp;drupal_static(__FUNCTION__, array());   if (!isset($controllers[$entity_type])) {     $type_info = entity_get_info($entity_type);     $class = $type_info['controller class'];     $controllers[$entity_type] = new $class($entity_type); // Line 7506   }   return $controllers[$entity_type]; }   However, I just don't understand Drupal at a low enough level to understand the relationships between these variables and functions.  I also searched Drupal for people with a similar problem:  (1)  who claims that I have to install the submodule "Relation Endpoints Field", but searching Drupal, .  (2) :  DELETE FROM cache; DELETE FROM cache_filter; DELETE FROM cache_menu; DELETE FROM cache_page; DELETE FROM watchdog;   (via , misleading name, I know.)  Clearing those tables from the cache did not help.  (3) And another user . Because I can't find a module specifically pertaining to old search facets, I went into MySQL system table, and marked the Search module's line with a 0 status, indicating to stop loading the module altogether.  On that same link as (3), one user suggested enabling the , so I dropped all the tables, imported the latest stable .sql, and installed the module, but after recreating the problem (trying to add a field to a content-type), that as well concluded fruitless.  Still, the problem remains  The Fatal error shows up on every page, making the site completely non-functional. Any help or direction would be immensely appreciated. Thank you for looking. To show: $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$  Now, the equation $x^{2} =3$ has a solution in $\mathbb{R}$, but not in $\mathbb{Q}$.  Hence they are not isomorphic to each other.  Is that right, or do I need to prove something else?  Thanks. How to add citation for a whole paragraph? If I write a report and a whole paragraph is based on one source but not cited verbally it seems unnecessary to include the reference after every sentence:  Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua (source 1). Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat (source 1).  Now I found a Phd thesis that puts the reference after the last dot of the paragraph to indicate that it is based on this very source:  Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. (source 1)  Is this a common practice? Is it generally understood? What are other ways to add the reference for a whole paragraph? Can I run Android apps on Ubuntu? Should we allow adding rep to bounties to extend their duration? There are a few variables here that I was thinking about:   Spend rep to extend a bounty for a certain amount of time Allow the rep spent to extend duration to add into the original bounty Create minimums on the amount necessary to extend dependent on how long it has been up   These are just ideas and can be tweaked as necessary, but the idea in whole is to encourage people to spend more on rep and avoid "wasted" rep because there wasn't a good answer to a question while minimizing the amount of possible abuse. This was part of the idea to spend a small amount of rep to extend duration without adding to the rep so that you wouldn't have to spend the whole amount over again for a couple more days.  The typical use case that I am thinking of is when you create a bounty and it still gets no attention. I think it would be cool if you could then add onto it to turn it into a higher bounty instead of having those points completely gone.  Possible abuses:   Keeping bounty questions alive for inappropriately long times. There is already an answer but the author does not want to award the bounty.   For 1, you can resolve this by increasing the amount necessary to keep a question alive exponentially similar to what is already in place. You could also halve the amount of extended duration per extension. For 2, this is a problem with the existing system as well, but the author does not have a positive incentive not to award the points since he himself still cannot get them back. Does the Planck scale imply that spacetime is discrete? How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$ I've used following codes and no luck  1  //Hide admin footer from admin function change_footer_admin () {     return ' '; } add_filter('admin_footer_text', 'change_footer_admin', 9999);  function change_footer_version() {     return ' '; } add_filter( 'update_footer', 'change_footer_version', 9999);   2  function wpbeginner_remove_version() { return ''; } add_filter('the_generator', 'wpbeginner_remove_version');   3  function my_footer_shh() {     remove_filter( 'update_footer', 'core_update_footer' );  } add_action( 'admin_menu', 'my_footer_shh' );   Still I see those texts and version on my footer.   So what is the accurate way to remove these? I've added these code snippets into my child theme function.php What do the various dpkg flags like 'ii' 'rc' mean? I frequently need to check which packages are installed, and I use the following command:  dpkg -l | grep foo  which gives the following output  Desired=Unknown/Install/Remove/Purge/Hold | Status=Not/Inst/Cfg-files/Unpacked/Failed-cfg/Half-inst/trig-aWait/Trig-pend |/ Err?=(none)/Reinst-required (Status,Err: uppercase=bad) ||/ Name                       Version         Description ii   foo                       &lt;version&gt;         &lt;description&gt;    What does the ii mean?  What other flags are there? How to read the flags? (because the explanation is quite complicated, IMO)   Thanks. The uniform limit of Continuous Functions is continuous.   Proof Let $\varepsilon &gt; 0$. There exists $N$ in natural numbers such that $n&gt;N$ implies  $|f_n(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. In particular $|f_{N+1}(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. Since $f_{N+1}$ is continuous at $x_0$ there is a $\delta&gt;0$ such that, $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f_{N+1}(x)-f_{N+1}(x_0)|&lt; \frac{\varepsilon}{3}$ Now we conclude $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f(x)-f(x_0)|&lt; 3 \cdot \frac{\varepsilon}{3} = \varepsilon$.  Can someone explain what is happening here? I don't follow it all. How can I check if a word is contained in another string using PHP? I'm currently working my way through Harvard's online abstract algebra lectures (if you're interested, you can find them ).  The lectures come complete with notes and homework problems.  Of course, since I don't actually go to Harvard, I can't hand in the homework assignments to find out if I'm doing them correctly.  So I've decided to try to crowd source the grading of my solutions.  I'll post individual questions as I finish them and wait for comments.  I would like to get critiques of not just my reasoning but the style of my write ups as well.  Also, any alternative approaches to the problem would be welcome.  I've looked for forums dedicated to these kinds of OCW courses but have not been able to find any.  This surprises me.  It seems like, with the advent of these free online educational resources, an online meeting place for those who take advantage of them would be a natural offshoot.  So this might be a bit of an experiment.  Unless I'm the 873rd person to post something like this here.  If that's the case, sorry for being so long winded.  Anyway, on with the question.  This one is assigned in the 4th lecture.     Let $V$ denote the Klein 4-group.  Show that $\operatorname{Aut}(V)$ is isomorphic to $S_3$.   (I've moved my solution below to keep this question from showing up in the unanswered list.)  Thanks...