suppose $f\in L^p(X,\mathscr{A},\mu), \forall p\in[1,\infty)$, then \begin{equation}   \| f\|_{\infty}=\lim_{p\rightarrow\infty}\| f\|_{p} \end{equation}  I can prove it on condition that $X$ is finite measure, and I think if not, it is not true, but I do not have a counterexample.   thanks
Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$?    I don't know where to start.
Any hard-working student is a counterexample to "all students are lazy".  What are some simple counterexamples to "if random variables $X$ and $Y$ are uncorrelated then they are independent"? they were question banned; it turned out it (in part) it was because of deleted questions.   was deleted by the Community user.  I'm aware it will delete old posts that have a negative score, but for a minute or two, I was really puzzled as to why it was unilaterally deleted without some notation left.  Can we have the Community user leave a notation in a post's history as to why the post was deleted?    Something like the following would suffice:     Post deleted automatically due to having negative votes and no activity for three months How to redirect output of an entire shell script within the script itself? Is it possible to redirect all of the output of a Bourne shell script to somewhere, but with shell commands inside the script itself? Redirecting the output of a single command is easy, but I want something more like this: #!/bin/sh if [ ! -t 0 ]; then     # redirect all of my output to a file here fi  # rest of script...  Meaning: if the script is run non-interactively (for example, cron), save off the output of everything to a file.  If run interactively from a shell, let the output go to stdout as usual. I want to do this for a script normally run by the FreeBSD periodic utility.  It's part of the daily run, which I don't normally care to see every day in email, so I don't have it sent.  However, if something inside this one particular script fails, that's important to me and I'd like to be able to capture and email the output of this one part of the daily jobs. Update: Joshua's answer is spot-on, but I also wanted to save and restore stdout and stderr around the entire script, which is done like this: # save stdout and stderr to file  # descriptors 3 and 4,  # then redirect them to &quot;foo&quot; exec 3&gt;&amp;1 4&gt;&amp;2 &gt;foo 2&gt;&amp;1  # ...  # restore stdout and stderr exec 1&gt;&amp;3 2&gt;&amp;4 Possible Duplicate:     Which one of these is correct?  It is I who am at fault. It is me who is at fault.  The word &quot;is&quot; is a conjugation of &quot;be&quot; which is a linking verb. I also want to know the same for 2nd person.  It is you who are at fault. It is you who is at fault. UNITY : I want to make my UI text fade in after 5 seconds I know this question has been asked and asked but none of the topics i visited gave me an answer. I'm making a game programmed in c# where you control a boat and i want my UI to appear smoothly by fading in after X seconds. May someone help me? (How) Can SE sites change their reputation payout? A recent question on  reminded me that there is a  about awarding 10 rep per question upvote. This is mainly due to the relative difficulty and usefulness of asking/answering being different on the site (or appearing to be, at least).  The highly voted answer on PPCG is "Yes, we should do that," but it's been well over three years since it was posted. The answer there said that (at least at that time) meta.so had this implemented already, so it wouldn't/shouldn't be hard to do.  Given that it's been quite some time, I don't know if that's (still) true. I know that meat has undergone changes since then, but as long as I've been actively participating, meta.se (and meta.so before the split) has given 5 points per question upvote, not 10.  My question is: What would the process to change it look like? It's been discussed on meta.ppcg and agreed upon, so what's the next step? Or is it not feasible at all? How do you find out which program is using too much memory? Ubuntu has been crashing on me recently. I think its because it runs out of memory so I ran the free -m command and found that my memory usage was really high. So then, i ran top to find the culprit, but the displayed processes were using less than 1.5% of memory. How do I know which program is making ubuntu crash/run out of memory? Below is the output:  shafee@shafee-pc:~$ free -m              total       used       free     shared    buffers     cached Mem:          3824       3714        110          0        978       1954 -/+ buffers/cache:        780       3044 Swap:           99          0         99  shafee@shafee-pc:~$ top top - 02:12:14 up  1:24,  2 users,  load average: 0.16, 0.42, 1.49 Tasks: 182 total,   1 running, 181 sleeping,   0 stopped,   0 zombie Cpu(s):  2.9%us,  1.9%sy,  0.3%ni, 79.3%id, 15.5%wa,  0.0%hi,  0.2%si,  0.0%st Mem:   3916708k total,  3803848k used,   112860k free,  1002308k buffers Swap:   102396k total,        0k used,   102396k free,  2001852k cached    PID USER      PR  NI  VIRT  RES  SHR S %CPU %MEM    TIME+  COMMAND              4200 root      20   0  289m  53m  38m S    2  1.4   1:06.45 Xorg                 5590 shafee    20   0 19348 1368  956 R    2  0.0   0:00.01 top                     1 root      20   0 24124 2136 1264 S    0  0.1   0:02.05 init                    2 root      20   0     0    0    0 S    0  0.0   0:00.00 kthreadd                3 root      20   0     0    0    0 S    0  0.0   0:24.23 ksoftirqd/0             6 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/0             7 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/1             9 root      20   0     0    0    0 S    0  0.0   0:00.11 ksoftirqd/1            11 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/2            13 root      20   0     0    0    0 S    0  0.0   0:03.89 ksoftirqd/2            14 root      RT   0     0    0    0 S    0  0.0   0:00.00 migration/3            16 root      20   0     0    0    0 S    0  0.0   0:00.16 ksoftirqd/3            17 root       0 -20     0    0    0 S    0  0.0   0:00.00 cpuset                 18 root       0 -20     0    0    0 S    0  0.0   0:00.00 khelper                19 root       0 -20     0    0    0 S    0  0.0   0:00.00 netns                  21 root      20   0     0    0    0 S    0  0.0   0:00.01 sync_supers         shafee@shafee-pc:~$   Or is it normal to have 110mb of free memory and is my issue related to something else? Why do some services list an IPO date that is well after historical price data you can find elsewhere? The IPO date I find for the MMM Corporation is 1/13/1978 ().   Google Finance has price data going back to that date ().   However, Yahoo Finance goes back to 1970 (), as well as other ticker sources.  The main concern, of course, being: How can there be published price information from before an IPO? Inaccurate altitude GPS data with my Android? What is the purpose of  in Python — especially with respect to when I would want to use it, and when not? Finding the volume of the tetrahedron. Find the volume of the tetrahedron with the vertices $P(1,1,1)$, $Q(1, 2, 3)$, $R(3, 1, 2)$, and $S(2, 3, 1)$. Can't install Ubuntu 18.10 on XPS 15 - EFI\BOOT\mmx64.efi not found I tried to install Ubuntu 18.10 on my XPS 15 9570 earlier. Everything was working fine until I got to the partition selection part of the installation. That's when the installer crashed and I had to shut down my machine. I think it's because I had my SATA configuration set to RAID ON instead of AHCI, which is now fixed.  Now when I try to run the installer from my bootable USB I get the following error  Failed to open \EFI\BOOT\mmx64.efi - Not Found Failed to load image \EFI\BOOT\mmx64.efi: Not Found Failed to start MokManager: Not Fond Something has gone seriously wrong: import_mok_state() failed   Hoping someone might have some idea as to what is going on How Special Relativity causes magnetism Hide c# windows application source code I wrote a windows application using C# .Net 2.0 and i want to do something which hide the source code, so when any one use refactor tool can't see the source code. I used dotfuscator but it just changed the function names but not all the source code.  UPDATE: I want to hide the source code, not because of hiding the key, but to hide how the code is working.  Thanks, I've been messing around with Unity and broke something, how do I "start over"? How to deserialize a JSON String to Apex $f:\mathbb{R}^3\to\mathbb{R}, f(rx)=r^{\alpha}f(x)$ for some $\alpha&gt;0$, $\forall x\in\mathbb{R}$, for any $r\in\mathbb{R}$, could any one tell me which of the following is true?  $1.$ If $f(x)=f(y)$ whenever $||x||=||y||=\beta&gt;0$, then $f(x)=\beta ||x||^{\alpha}$  $2.$ If $f(x)=f(y)$ whenever $||x||=||y||=1$, then $f(x)= ||x||^{\alpha}$  $3.$ If $f(x)=f(y)$ whenever $||x||=||y||=1$, then $f(x)=c||x||^{\alpha}$ for some constant $c$  $4.$ If $f(x)=f(y)$ whenever $||x||=||y||$, then $f(x)$ must be a constant function. Thank you. Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k&gt;0$ is constant. Show that $f$ is a polynomial of degree at most $n$. Kontact in Kubuntu 16.10 will not start because the Akonadi server will not start Root of unity for $n+1$ Let $n \ge 1$ and $u_{1},u_{2},...,u_{n}$ are all root of unity for $n+1$ and there are different from $1$.(a) Find polynomial $f \in \mathbb R[x]$ of degree $n$ for which: $f(u_{k})=0$ for $k=1,2,...,n$.(b) Calculate real coefficient and imaginary part for $(2-u_{1})(2-u_{2})\cdot...\cdot(2-u_{n}).$    I have a knowledge about root of unity but I completely don't know how to do it because I think that I need some extra trick to do it but my ideas finished. I know that $\varepsilon_n^{(k)} = \cos\left(\tfrac{2k\pi}{n}\right) + i\sin\left(\tfrac{2k\pi}{n}\right) = e^\frac{2\pi i k}{n}$ and for $n&gt;1$ $\sum _{{k=0}}^{{n-1}}e^{{\frac  {2\pi ik}{n}}}=0$ but I think it is too little knowledge to solve this task.Can you help me?