Assume $f$ is Riemann integrable and further assume that $\int_a^x f=0$ for all $x$. How would I go about showing that $f$ itself is $0$ almost everywhere?  I am new to Lebesgue's measure theory so I am hoping for a somewhat elementary proof if possible?  I know that almost everywhere means all except a set of measure zero. I was wondering if I could get a point to start on? We are told $f$ is Riemann integrable, so that means by Lebesgue's criterion for Riemann integration that there are at most countably infinitely many discontinuities. Thank you!
I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some sequence of functions converges weakly to zero in $L^2$.   I managed to show (with some help) that the limit $f$ (of a subsequence) satisfies $\int_0^x f \ dm=0$ for all $x&gt;0 $. From this I want to conclude that $f=0$ a.e. I can do this with the fundamental theorem of calculus in its Lebesgue version but there ought to be a more elementary proof.  Can someone here help me out?
How To: Execute command line in C#, get STD OUT results Using Gnome 3.18. I share files between other family members, but the default umask on my distro (archlinux) is 0022. So every file/directory created is not writable for our common group.  I tried to put umask 0002 in /etc/profile but the gnome session is still using 0022. It's working for a login bash shell, though.  I also tried to add this line in /etc/pam.d/system-auth:  session   required  pam_umask.so umask=0002  It has the same effect as the one in /etc/profile. I tried  If I change the umask manually in a gnome-terminal shell, then I launch an application from it, say gedit, then the files created by it have the wanted permissions. If I launch gedit from the gnome menus, it doesn't. So my matter is really to set the umask for the gnome session, and I can't find where to do it.  EDIT (to answer Gilles' comment): I'm using gdm 3.18 as the DM. I also tried to add the pam_umask line into /etc/pam.d/gdm-launch-environment. All other gdm-* files contains includes of session from the system-auth file, so they should not need more. It doesn't change anything.  /etc/login.defs contains UMASK 077 but also USERGROUPS_ENAB yes which should set the umask to either 0077 or 0007 for users whose primary group is the username.  The only file that contains 022 for umask in /etc is /etc/profile but that was my first try.  As for /etc/Xsession.d, I don't have this directory. Besides, as wayland is now the default display server, I'm not sure the umask should be set as part of X initialisation, even if I'm still using it myself. What is accountsd and why does it want to use the "login" keychain? What's the past participle form of the word exit? Is it exit (irregular, like set)? exited? exitted? On one page I found exited but if that's the case why isn't it exitted (double t) like with the word emit - emitted? Is there a rule when the consonant at the end is doubled and when not? I have the following code in my theme's archive.php:  &lt;?php the_archive_title( '&lt;h1 class="page-title"&gt;', '&lt;/h1&gt;' ); ?&gt;   This gives me titles like "Category: Russia", "Tag: America", "Author: John".  I would like to remove the "Category:", "Tag:" and "Author:" part and just display the category, tag and author names.  Does anyone know how to accomplish this?  Thank you. Cannot dismiss scrollbar in code box, blocks view This question is a duplicate of  (but this question can't be marked as a duplicate because that question is on meta.stackoverflow.com, and this question is on meta.stackexchange.com).  I'm seeing a bug on stackoverflow.com. I'm not sure if the culprit is Stack Overflow, Google Chrome, or macOS.  (Below I refer to a "code box". By that I mean some text that begins with 4 spaces when editing a question or answer.)  When I hover over a code box that contains one long line of code, a horizontal scrollbar appears. When I stop hovering over the code box, very frequently (maybe 3 out of 4 times) the scrollbar does not disappear. Clicking elsewhere on the page also does not dismiss the scrollbar. The only way to dismiss it is to refresh the page.  ...After experimenting more, it appears to always happen if I enter the box from below, but not always when I enter from above. It seems to be triggered by hovering over the lower half of the box.  Also if I enter from the top and scroll to the right, then until I refresh the page again that scrollbox will behave normally (only show a scrollbar when scrolling, and disappear after). In that case the scrollbar also has a different appearance, being much less tall (maybe a quarter of the height of the code box rather than half as tall).  Further, if I get a box and scrollbar into that functional state and then hover over the lower part of the box for a bit, then sometimes eventually it will break and stay broken until refresh again.  And in any case hovering over the lower part of the box causes the scrollbar to get taller.  Here is a screenshot of the issue:        Google Chrome 71.0.3578.98 (Official Build) (64-bit)  macOS High Sierra 10.13.6  Here is a code box that repros the issue for me:  fooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo   Here is another:  foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo QGIS - export a project with dependencies Find the point which is furthest away from existing points while remaining within the convex hull Eclipse doesn't start on Ubuntu 18.04 I installed eclipse via Ubuntu Software. When I start it, it take some seconds and then comes an error window showing the path of the error.  The following is the .log file:  !SESSION Wed May 02 12:53:28 BRT 2018 ------------------------------------------ !ENTRY org.eclipse.equinox.launcher 4 0 2018-05-02 12:53:28.587 !MESSAGE Exception launching the Eclipse Platform: !STACK java.lang.ClassNotFoundException: org.eclipse.core.runtime.adaptor.EclipseStarter     at java.base/java.net.URLClassLoader.findClass(URLClassLoader.java:466)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:566)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:499)     at org.eclipse.equinox.launcher.Main.invokeFramework(Main.java:626)     at org.eclipse.equinox.launcher.Main.basicRun(Main.java:584)     at org.eclipse.equinox.launcher.Main.run(Main.java:1438)     at org.eclipse.equinox.launcher.Main.main(Main.java:1414)   What is the problem and how can I fix it? Where $F$,$K$ are two fields. What does the element in the composite field $FK$ look like? All the elements are generated by the elements of $F$ and $K$? (combination of the elements of $F$ and $K$) I think $FK$ is pretty close to the free product of $F$ and $K$; is $FK \subsetneq F*K$? Thank you! CSS: Special Fluid Layout Problems Energy-momentum tensor transformation I've been trying to find how the energy-momentum tensor changes if we add a total derivative to the lagrangian: $$L\to L+\mathrm d_\mu X^\mu.\tag{1}$$   From the answer key:  $$T^{\mu\nu}\to T^{\mu\nu}+\partial_\lambda F^{\lambda\mu\nu}\tag{2}$$ Where $$F^{\lambda\mu\nu}=\frac{\partial X^\lambda}{\partial(\partial_\mu\phi)}\partial^\nu\phi -\frac{\partial X^\mu}{\partial(\partial_\lambda\phi)}\partial^\nu\phi.\tag{3}$$ Is the above answer right? I'm not able to get this result! any help? Accidentally ejected External drive while downloading file, now having problems reading I'm hoping someone can help! I recently downloaded a torrent directly onto an external hard drive, (using a 2009 MacBook Pro and 1TB Seagate Expansion portable Drive) somehow during the download the drive was accidentally ejected, since this incident when I plug the drive back in it’s having problems reading.  The drive works fine for about 10 minutes, I can read/copy/paste and access files but then starts to wreak havoc; apps hangs, it hangs the Finder, etc.  I’ve tried repairing it in Disk Utility, which reads and repairs this disk but problems inevitably continue as described above.  I’ve deleted the Torrent file also. All my files on this external drive so re-formatting is not an option until I can retrieve them. I'm a software developer and as such I build and run programs hundreds of times per day. I've noticed that since I started using Catalina, after compiling and linking there is a delay (which can be seconds) when the internet is slow or having problems (I have bad internet often). If I disable wifi and rebuild, the delay goes away. How can I disable this so my builds run faster? How did the AT-ATs get to the surface of Hoth? A reflection question about a complex function First axiom of sheaves: in noetherian topological spaces the direct limit presheaf is a sheaf. What is the word that means “that which can be fixed up at home”?   For example, I buy parts from a store, bring the parts home, and then can construct the thing out of those parts by myself. What is the word that means such an object, one that can be made at home in a do-it-yourself style? Consider this situation:  A ball is moving forward and undergoing rotation. Assume that it is not slipping. Eventually, the velocity and rate of rotation of the ball decrease, and it comes to a halt. But if you observe the direction of friction (when the ball is rotating clockwise), you will see that the friction should have provided a clockwise torque to the ball and the angular velocity of the body should have increased. But this doesn't happen. Why? URL line breaks with biblatex How do I allow long URLs to have line breaks at any point. This is in the bibliography not the main text. I am trying to save space so don't want extra spaces to be added to make it only break at slashes, for example, as it does by default.  I am using biblatex+biber.