Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$?   Clearly this does not hold for $p = \infty$, since given functions with same hight, pointwise convergence does not imply uniform convergence.  And when $p=1$, it becomes Scheffe's lemma.  For $ 1&lt; p &lt;\infty$,  we have   $$\int |f_n - f|^p dm \leq 2^{p-1}\bigg(\int  |f_n|^p + |f|^p dm\bigg). $$ Using General Lebesgue Dominated Convergent theorem, we have  $$\lim_n \int |f_n - f|^p dm = \int \lim_n |f_n - f|^p dm,$$ if and only if  $$\lim_n \int |f_n|^p + |f|^p dm = \int \lim_n |f_n|^p + |f|^p dm$$ which holds from the assumption that $||f_n||_p \rightarrow ||f||_p$.   Is my argument correct? Thanks!
Let $1\leq p &lt; \infty$. Suppose that    $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite),  $f_k \to f$ almost everywhere, and  $\|f_k\|_{L^p} \to \|f\|_{L^p}$.    Why is it the case that $$\|f_k - f\|_{L^p} \to 0?$$   A statement in the other direction (i.e. $\|f_k - f\|_{L^p} \to 0 \Rightarrow \|f_k\|_{L^p} \to \|f\|_{L^p}$ ) follows pretty easily and is the one that I've seen most of the time. I'm not how to show the result above though.
How do I use the Natural Earth SQLite DB with QGIS? I just downloaded the Natural Earth data in SQLite format from . I had assumed that this would be a SpatiaLite database, but it appears not to be! QGIS is failing to recognize it as a spatial database. OGR supposedly supports reading geometries even when stored in a plain SQLite database, but maybe QGIS doesn't use OGR for SpatiaLite?  The SQLite database does have geometry_columns and spatial_ref_sys tables. Is there some way to convert it to a full SpatiaLite database? I may not be coming in tomorrow...    I might not be coming in tomorrow...       When should I use "may" and "might"? When to throw an exception? Sending Email via gmail smtp server in JAVA What are the grammatical rules determining whether to use which or what?  I usually go by intuition. It's "What car are you looking at?" and "What bicycle do you like most?" but "Which song did you sing?" Quite often, I don't know whether to use what or which.  How can I tell? After some searching on stack / google I decided to ask this here, this is the second time I'm facing this issue this time with blender 2.8, previously with 2.7 and without a clue to how I got here.  Some of my objects appear hidden(10/12), and I can't seem to find a way to unhide them.  Solutions to the most common rookie mistakes seems to be   In local-view solved by pressing numpad / hidden by pressing H or Shift + H, unhiding with Alt + H   I'm not in local view, I can toggle visibility for 2/12 objects with Shift + H &amp; Alt + H, but the rest stays gray in the collection.  Last time I solved this by exporting the whole model as STL and importing it back again, I then had to re-group all the pieces but there must be a more simple solution.  Here's the .blend if someone wants to have a go at it: Find the equation of the plane knowing that it passes through 3 points How to layout irregular paragraph shape I have found this in the cover of seminar(one tool to make slide) document.  The final result I have captured below:    The complete document can download .  I wonder how to make this. Is there any easy way? If $r,s,t$ are prime numbers and $p,q$ positive integers s.t. $\text{lcm}(p,q)$ is $r^2s^4t^2$, then the number of ordered pairs $(p,q)$ is? If $r,s,t$ are prime numbers and $p,q$ are the positive integers such that LCM of $p,q$ is $r^2s^4t^2$, then the number of ordered pairs $(p,q)$ is?  My attempt to the solution:  Let $r^a s^b t^c$ be the LCM where $a=2,b=4,c=2$  Then the cases that arise are  I case   1) $p$ has $r^as^bt^c$ then there are a total of $(2+1)(4+1)(2+1)$ options  II case   1) $p$ has $r^as^b$ or $s^bt^c$,$r^at^c$ then there are total of $(2+1)(4+1)(2+1)(3)$ options.  The answers comes out to be $45*4$ but the answer is $45*5$. Which case am I missing? Remove "Show Desktop" button next to the clock in Windows 7 I have the following configuration, but I have not able to find any documentation on how to set a maximum backup files on date rolling style.  I know that you can do this with size rolling style by using the maxSizeRollBackups.  &lt;appender name="AppLogFileAppender" type="log4net.Appender.RollingFileAppender"&gt;     &lt;file value="mylog.log" /&gt;     &lt;appendToFile value="true" /&gt;     &lt;lockingModel type="log4net.Appender.FileAppender+MinimalLock" /&gt;     &lt;rollingStyle value="Date" /&gt;     &lt;datePattern value=".yyMMdd.'log'" /&gt;     &lt;layout type="log4net.Layout.PatternLayout"&gt;       &lt;conversionPattern value="%d %-5p %c - %m%n"  /&gt;     &lt;/layout&gt; &lt;/appender&gt; Find out which process is locking a file or folder in Windows Is there an MD5 Fixed Point where md5(x) == x? Is there a fixed point in the MD5 transformation, i.e. does there exist x such that md5(x) == x? What is a reliable method to locate floating islands in Terraria? I have not had to mess with mailto links much.  However I now need to add a link in the body of a mailto if it is possible.    Is there a way to add a link or to change the email opened to an html email vs a text email?  Something like:  &lt;a href="mailto:test@test.test?body=The message's first paragraph.%0A%0aSecond paragraph.%0A%0AThird Paragraph.%0A%0ALink goes here"&gt;Link text goes here&lt;/a&gt; Some common problems with displaying routes created from street data, such as bus routes, include:   Routes may overlap themselves, such as when a route takes the same road in opposite directions Routes may overlap other routes that take the same street. Routes are often overly detailed, or "noisy", owing to their derivation from street data. Indicating a route's direction unambiguously without overloading it with arrows can be a challenge.   What are some techniques for dealing with these issues in order to create a practical and usable route map, such as this ? Do you know of any such public transport maps that were produced by automatic techniques rather than manual illustration/digitization?  Specifically I am using ArcGIS 10, but solutions involving other software are welcome as well.  One technique I have had some success with for dealing with 1) is using ArcMap's  with a negative offset -- assuming the lines are oriented in the direction of travel, this will offset the lines such that the right and left-hand sides are distinct and oriented as you would expect (at least for  countries, for left-hand traffic countries one would use a positive offset). However the result tends to be a bit wonky around turns, interchanges and and U-turns, and is inflexible when dealing with multiple routes.  Another possibility that I am not very familiar with but which seems promising is  -- however, I have not had much luck in finding examples of using representations for transportation maps and whether it might fit the bill. Windows 7 / 8 stop modal dialogue flash on parent focus Is there some setting in Windows where I can change the default behavior that causes the border and title of a modal dialogue box to flash like 6 or 7 times when I click on the parent window?  Simply focusing it would be plenty to let me know what I did was wrong.   I don't feel the need for all the sirens and bull horns and strobe lights to let me know I clicked in the wrong place. What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? The title pretty much says it all.  Of course, one answer (IMO unsatisfactory) to such questions goes something like "a definition is a definition, period."  In my experience, mathematical definitions are rarely completely arbitrary.  Therefore I figure there must be a good reason for insisting that a field be a non-trivial ring (among other things), but it's not obvious to me.  I realize that a trivial ring would make for a very boring field, but it is also a rather boring ring, and yet it is not disallowed as such.  EDIT: I found a hint of a rationale in the statement that "the zero ring ... does not behave like a finite field," but I could not find in the  for this assertion exactly how the zero ring fails to behave like a finite field. Why can Naruto still use Rasenshuriken? When Naruto first used Rasenshuriken against Kakuzu the effect was massive. He destroyed   two of Kakuzu's hearts, but there were a huge after effect to him. If I'm not mistaken, Tsunade advised Kakashi to ask Naruto never to use the technique again due to the huge risk. So why is Naruto still able to use the technique and have no side effects after that battle? I may have missed out on something. Exclamation(!) turned in to question mark(?) in title field I just submitted a question () with an exclamation (!) at the end of the title (with one space between the word and the exclamation) but it turned in to question mark (?).   I removed the space between the word and the exclamation, then it shows normal.   Is this an intended feature?