I've encountered two similar statements that I don't know how to prove.  $1.$ let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. let $f:X\to [0,\infty)$ be measurable. Show that:  $\int_X fd\mu =\int_{[0,\infty)}\mu(\{x|f(x)&gt;t\})dm(t)$  $2.$ let $(X,\Sigma,\mu)$ be a complete measure space. let $f\in L^1(\mu)$ be non-negative. Show that:  $\int_X f d\mu=\int_{[0,\infty)}\mu(\{x|f(x)\ge t\})dm(t)$  I've tried to solve it but i'm not even sure what is $m(t)$ and how do I integrate over  $\int_{[0,\infty)}\mu(\{x|\chi_A&gt;t\})dm(t)$,  for $f=\chi_A$  how do I continue?
Let $({\Bbb R},{\mathcal A},\mu)$ be the measure space where ${\mu}$ is the Lebesgue measure.  Assume that $\int_{\Bbb R}|f|^pd\mu&lt;\infty$ ($p\geq1$). There is an exercise for proving that       $$\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s).$$   If one formally let $s=t^p$, then $ds=pt^{p-1}dt$ and one would immediately get the equality above. However, how can I justify it rigorously in the sense of Lebesgue integration?    [Edited] What puzzles me is that I have the "change of variable" theorem for Riemann integration, but all the integrations here are in the Lebesgue sense. Unless one has shown that the integrands are also Riemann integrable, how can I directly use "change of variable"?
I am flying from Los Angeles to Bangalore. I have a flight option from LAX to Minneapolis and from there to Paris. I have only 45 minutes at Minneapolis before my next flight. Is this too risky a schedule? I am not an American, so I guess I would need to cross emigration too? Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ I can't crack this one.      Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that  $a,b,c,z \in \mathbb{Z}$ and $c\neq 0)$ I have auto traced a photograph in illustrator - its got thousands of vectors once I expand it and illustrator has converted each of them to a filled colour - I want to now trace the edges of them with a grungy brush but using the underlying fill colour. Where in illustrator can I stroke or outline these fills using the underlying colour - surely this is something that the program automatically does?   I've found this script online but don't know how to implement it. I'm new to this so any help appreciated. Many thanks.   James   var idoc = app.activeDocument;   var sel = idoc.selection;    for (i=0; i&lt;sel.length; i++) {       var ipath = sel[i];       ipath.stroked = true;       ipath.strokeColor = ipath.fillColor;   } how to get the original entity on hook_entity_presave? i actually need it for the content_moderation_state entity   function hook_entity_presave(Drupal\Core\Entity\EntityInterface $entity) { if ($entity-&gt;getEntityTypeId() !== 'content_moderation_state' ) {     return; }     $moderation_state = $entity-&gt;get('moderation_state')-&gt;getString();   //code to get the original entity moderation state I am new to Stack  Exchange and I think it's very impressive.  One thing bothers me is that because it is so far down the line, I don't think beginners stand a chance of growing.  As time goes on, more and more questions are being answered and therefore if someone asks something with a really basic answer you can just comment. However, beginners can't comment until they have a reputation of 50. The only way they can build up is by answering questions. But if they answer a basic question long winded, they will be voted down because not so much effort was needed or because its already a duplicate and will lose the bit of reputation they have earned.  Can you tell me the future for beginners? The three lengths \textwidth, \linewidth and \hsize seem to all hold the width of the current line. At least I never saw some real difference between \textwidth and \linewidth in my code. Both seem to be identical even when changed, e.g. by a minipage environment.  Could someone give me an elaborate explanation of the differences between these lengths? I like to understand them in detail. I know that the first two are from LaTeX and the latter one from plainTeX.  There is also \columnwidth which seems to be identical to \linewidth or \textwidth in one-column mode, right? Except when the line is reduced by a minipage or quote etc. environment. querying Recently account viewed Anti-Matter Black Holes Assuming for a second that there were a pocket of anti matter somewhere sufficiently large to form all the type of object we can see forming from normal matter - then one of these objects would be a black hole.  Question is, would there be a difference between an anti matter black hole and a normal matter black hole - in terms of how would the matter/anti-matter make the black hole different, or would they be the same?  I would expect the answer to be that the black hole formed from anti matter would retain the anti-matter properties in such a way that if it was to merge with a black hole at same size formed from normal matter that they would annihilate  each other con convert into pure energy - Would that be a correct understanding? I had to take ownership of the special folder C:\Program Files\WindowsApps to fix . The issue is now resolved, so I'd like to reset the permissions to the way they were before I took ownership of the folder.  I reverted NT SERVICE\TrustedInstaller to the owner and removed my account's permission entries:     But I still have full control of the folder and can browse the folder in Explorer without getting the usual warning that I would have to take ownership:     Is it possible to restore the default permissions of this folder? How to add D-link dwm-156 to usb switch mode for automatic eject of driver CD / switch to modem mode? I am running Ubuntu 13.10, 32-bit.  I had every time need to go terminal to "eject" the windows driver CD,nor gone to disk &amp; eject windows driver CD to make the d-link dwm-156 gsm modem functioning through network manager.  It seems that usb mode switch is not working.  But on my friend's computer with ubuntu 14.04 ,usb mode switch is working and i can connect d-link dwm-156 to mobile broadband through network manager as soon as i get in.  Is there any way to make usb mode switch to eject the windows driver CD of d-link dwm-156 in my ubuntu 13.10,32-bit at start up?   The lsusb output is Bus 002 Device 013: ID 2001:a706 D-Link Corp before ejecting the CD (modem as storage device of windows driver).  The lsusb output is Bus 002 Device 005: ID 2001:7d01 D-Link Corp after ejecting the CD (as modem).   I don't want to use the driver in the CD,as i need to send sms &amp; find the data usage. Do MOSFETs have a diode built into them? I have noticed that a MOSFET symbol has a little diode in it (or at least what looks like a diode). Does this mean I do not have to worry about using a diode in a circuit that runs a motor using one? I would have used a diode in order to prevent reverse voltage from turning the motor by hand. Debugging errors of SAGA interpolation algorithms in QGIS Processing? I encounter recurrent problems with the geoalgorithms that perform interpolations. No matter what technique I use I end with the following errors. As stated in the OGR tab QGIS 2.0 is unable to open a file in C drive. I have very little IT background so the message error is meaningless.   Is there anybody able to decrypt the message and solve the problem?      here this is the algorithm i used For gravity, we have    General Relativity, which is a geometric theory for gravitation.  Is there a similar analog for Electromagnetism? This morning I started my Google Chrome browser as I do every day. The text size was suddenly bigger than normal, on all sites. I did not change any setting anywhere.  If I set the zoom to 90% it looks fine, but this breaks various things (e.g. Stack Exchange chat) so prefer to avoid it if possible.  Setting font size to Small in Chrome Settings also help, but it's too small.  I have uninstalled Chrome (deleting local data) and installed again, same thing. Tried beta channel as well, no change.  Windows 7 OS 64 bit, Display set to 125% from the very beginning. (Did not change this for years)  What could have caused this and how can it be solved?  This very question turns out to be a good example to demonstrate what I mean:        While the body itself is small (as I changed the font size in Settings), other things are still huge and not affected by the setting. Question 1: Let $x,y \in S$ such that $x\sim y$ if $x^2 =y^2\pmod6 $. Show that $\sim$ is an equivalence relation.    This is what I tried: Reflexive: $x^2\pmod6 = x^2$  implying $x\sim x$ Symmetry: suppose $x\sim y$, then  $x^2 =y^2\pmod6$ Operating by inverse of $x^2$ both sides we have $e=(x^2)^{-1} y^2\pmod6$ Then I have $y^2=x^2\pmod6$, hence $y\sim x$ I would like to get some corrections up to here because am not sure with this     Question 2. Find the orbits of the given permutation:    $$v:\mathbb{Z}\to\mathbb{Z}\quad\text{ defined by }\quad v(n)=n+3$$   Here is my solution which I would also want somebody to correct me! Let the relation $\sim$ be defined on $\mathbb{Z}$ by $a\sim b$ if $b=a+3$ I consider that the orbits of $v$ are the classes of $\mathbb{Z}$ with respect to $\sim$ given by  $\{b\mid b=a+3\}$. Thus I get only three orbits which are $\{1,4,7,10,…,3n-2\mid n\in \mathbb{Z}\}$, $\{2,5,8,…,3n-1\mid n\in \mathbb{Z}\}$, and $\{3,6,9,…,3n \mid n\in \mathbb{Z}\}$. bash: /usr/bin/locale-check: No such file or directory error after i downgraded from ubuntu 18.10 to 18.04 This is probably a silly question, but how can I run ANOVA using only the mean, standard deviation, and the sample size?  i.e.  Group  mean    sd    n  1:    200      5    5    2:    100     22    5  3:    150     14    5   I am using Statsoft Statistica software.  I know how to run it using the spreadsheet with all the data, but I have never done this using only the information listed above.  Any information would be greatly appreciated. Why are there so many different ways to measure disk usage? When I sum up the sizes of my files, I get one figure. If I run du, I get another figure. If I run du on all the files on my partition, it doesn't match what df claims is used. Why are there so many different figures for the total size of my files? Can't computers add?  Speaking of adding: when I add the “Used” and “Available” columns of df, I don't get the total figure. And that total figure is smaller than the size of my partition. And if I add up my partition sizes I don't get my disk size! What gives? What is the best way to explain how to choose between "its" and "it's"? Probably one of the most frequent grammar mistakes in the English language is:     The dog sat on it's mat.   Since spelling checkers don't catch it, and it is even logical, since you would correctly write:     The dog sat on Fluffy's mat.   What is the best way to explain to a learner of English how to choose between it's and its? If a company is bought by another company, does my stock rollover to the new company when it is combined with the new parent company?