To show that Egoroff's Theorem continues to hold if the convergence is point-wise a.e. and  $f$ is finite a.e. on  $E$.  And Egoroff's Theorem states that if $E$ is a set of finite measure and $\{f_n\}$ is a sequence of measurable functions that converges pointwise on $E$ to the real valued function $f$, then there exists a closed set $F$ that is roughly the same size as $E$ on which $\{f_n\}$ converges uniformly.
The following question is taken from Royden's Real Analysis $4$th edition, Chapter $3,$ question $28,$ page $67:$     Question: Show that Egoroff's Theorem continues to hold if the convergence is pointwise a.e. and $f$ is finite a.e., that is.       Assume $E$ has finite measure. Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise almost everywhere on $E$ to the real-valued function $f$ which is finite almost everywhere. Then for each $\varepsilon&gt;0,$ there is a closed set $F$ contained in $E$ for which    $$\{f_n\}\to f \text{ uniformly on }F \text{ and }m(E\setminus F)&lt;\varepsilon.$$   My attempt:   Let $A = \{ x\in E: f_n\not\to f \text{ pointwise} \}$ and $B=\{ x\in E: |f|=\infty \}.$  By assumption,  $$m(A)=m(B)=0.$$ So $A$ and $B$ are measurable.  Note that  $$m[E\setminus (A\cup B)] = m(E) - m(A\cup B) = m(E).$$ Fix $\varepsilon&gt;0.$ Since $\{f_n\}$ converges to $f$ pointwise on $E\setminus (A\cup B)$ to the real-valued function $f,$ by Egoroff's Theorem, there exists a closed set $F$ contained in $(E\setminus (A\cup B))\subseteq E$ for which  $$f_n\to f \text{ uniformly on } F \text{ and }m[(E\setminus(A\cup B)) \setminus F] &lt; \varepsilon.$$ Since $E$ has finite measure and $F\subseteq E,$ by monotonicity,  $F$ has finite measure. As $F$ is closed, it is also measurable.  Therefore, by Excision property, we have $$m(E\setminus F) = m(E) - m(F) = m[E\setminus(A\cup B)] - m(F) = m[(E\setminus(A\cup B)) \setminus F]&lt;\varepsilon.$$  Is my proof correct? I ask for verification because in this  Kenny's comment about countable union of null sets. I do not use this anywhere in my proof. So I wonder whether my proof miss out something.
Showing a tensor is not a simple tensor Let $(e_1,e_2)$ be the standard basis of $\mathbb{R}^2$. Why can the element $e_1\otimes e_2 + e_2\otimes e_1$ in $ \mathbb{R}^2\otimes_{\mathbb{R}} \mathbb{R}^2 $ not be written as a simple tensor $v\otimes w$ for $v,w\in \mathbb{R}^2$?  If we suppose $e_1\otimes e_2 + e_2\otimes e_1=v\otimes w$, then can we conclude that $e_1=\lambda e_2$ for some $\lambda \in \mathbb{R}$? This would give a contradiction since $e_1$ and $e_2$ are linearly independent. However, I am not sure if the argument is valid. Why do we say that the Earth moves around the Sun? In history we are taught that the Catholic Church was wrong, because the Sun does not move around the Earth, instead the Earth moves around the Sun.  But then in physics we learn that movement is relative, and it depends on the reference point that we choose.  Wouldn't the Sun (and the whole universe) move around the Earth if I place my reference point on Earth?  Was movement considered absolute in physics back then? How does the current limiting resistor for an LED affect current and voltage drops? I'm having a bit of trouble understanding currrent limiting resistors in simple LED circuits. I know that I can determine the optimal resistor like so:  \$\displaystyle R=\frac{V_{s}-V_{f}}{I_{f}}\$  But I'm having a hard time understanding how this one value modifies the voltage and the current to the correct values for the LED. For example, if my calcuations for a super bright blue LED (with \$V_{f}\$ being 3.0-3.4&nbsp;V and \$I_{f}\$ being 80&nbsp;mA, and a voltage source of 5&nbsp;V) gives me 25&nbsp;ohms (using the lower bound of the forward voltage), that's fine. So the current throughout should be 80&nbsp;mA and the voltage drop for the resistor and LED should be 2 and 3 volts, respectively.  But what if I used a 100&nbsp;ohm resistor instead? Or any other value—how would I calculate the voltage drops and current? Would I assume one of them stays the same? Start Ubuntu Unity With Dash Open How to escape quotes in shell? This is a question that has been addressed several times, not only here, but also in other sites of the stack exchange network (e.g. ). However, I cannot make any of the solutions work for me.  I'm running a command through ssh. Whenever I exit ssh, I want the command to die as well. This command is a daemon called ktserver that runs indefinitely until you press Ctrl-C.  I run it as follows: ssh -t compute-0-1 ktserver and, indeed, when I press Ctrl-C, the process ends gracefully and the ssh session ends.  However, if instead of pressing Ctrl-C, I kill the ssh process using the kill command (for example, sending SIGINT or SIGHUP), the ktserver process stays alive.  How can I make the ktserver always die independent on how ssh is killed?  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Note: The original question was to prove the more general     $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$$   for an integer $l$, which some of the answers address. Out of Bag Error makes CV unnecessary in Random Forests? How do I color intersection of two ellipses in different color. For instances, the ellipses should themselves be colored as yellow and their intersection as red. My code for drawing ellipses is given below  \draw (-4,0) ellipse (8 and 3); \draw (8,0) ellipse (8 and 3);   UPDATE: In the comments to individual answers, I have requested to show it for three ellipses. I want to write a function that takes an array of letters as an argument and a number of those letters to select.   Say you provide an array of 8 letters and want to select 3 letters from that. Then you should get:  8! / ((8 - 3)! * 3!) = 56   Arrays (or words) in return consisting of 3 letters each. Difference between == and === in JavaScript What is the difference between == and === in JavaScript?  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[A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).][1]  If $n = 5$ then  $$\left\lfloor1\sqrt{2}\right\rfloor+ \left\lfloor2\sqrt{2}\right\rfloor + \left\lfloor3\sqrt{2}\right\rfloor +\left\lfloor4 \sqrt{2}\right\rfloor+ \left\lfloor5\sqrt{2}\right\rfloor = 1+2+4+5+7 = 19$$  Sequence from $1$ to $20$ is:  $S=\{1,2,4,5,7,8,9,11,12,14,15,16,18,19,21,22,24,25,26,28\}$  I want to find answer for $10^{100} n$? Load external php library in custom module I’ve read a lot of topic on this but i can’t figure why it is not working for me.  I’ve built a custom module. It is working fine since return $output; in my controller class is working : I can see in front-end a page with my custom html.  I’ve written a script outside Drupal in order to test. Basically, i connect to a webervice and retrieve all the datas. It’s working fine.  Now, i want to execute that script inside the controller class.  I need to load an external php library to instantiate an object.  In my test php file (outside Drupal) a simple require_once is working.  require_once('nusoap.php');   But it is not working in my controller class.  So, i’ve activated the library module and do :  $path = libraries_get_path('nusoap') . DRUPAL_ROOT . '/modules/contrib/webservices_tr/webservices_dc/src/Controller/nusoap.php';  require($path);   It’s still not working.   Error message indiquactes :     Error: Class 'Drupal\webservices_dc\nusoap_client' not found   I’m sur the file path is correct because if i do this, i can see the content of the file.  echo file_get_contents(DRUPAL_ROOT . '/modules/contrib/webservices_tr/webservices_dc/src/Controller/nusoap.php'); die();   I’ve also tried this :  in nusoap.php (where there is the class i need to include),    namespace Drupal\webservices_dc;   and in module controller class :  use Drupal\webservices_dc\nusoap_client;   But it is still not working.  Finally, i’ve copied the content of the external php library inside the module controller and it is working.  So i’m nearly sure the issue is related to the loading of that external library.  What is the correct way to load external php library (and wait it is fully loaded) in order to use it in a custom module ? How to read from two input files using while loop Why does a subpanel need separate ground and neutral? Why does a power sub panel have to have separate neutral and ground bars other than code satisfaction?  Does a sub panel need a main breaker if there is one at the main?   Also, does the sub panel need a ground wire connected back to the main panel in the same building? I am currently developing a website which is going to take some time to finish. I need to move some 'stuff' to production frequently, while maintaining a under construction status. My front page will clearly mention that my website is under construction.  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I can see that this looks a little egoistic.  : One devoted to one's own interests and advancement.