Suppose $B$ is a Borel set and $f$ mapping  from $B$ to $R$ is a non decreasing function.Prove that $f(B)$ is a Borel set. f is a Borel measurable function because the inverse image of $(a, \infty)$ will be of the form $(t, \infty)$ or $[t,\infty)\cap B$ which is a Borel set. I cannot define inverse function because it may not be strictly increasing. How do I proceed?
I'm studying measure theory and two question came to my mind:        If $f:\mathbb{R}\to\mathbb{R}$ is monotone and $B\subseteq\mathbb{R}$ is borel, is the image $f(B)$ borel?   If $f:\mathbb{R}\to\mathbb{R}$ is a monotone function (say, non-decreasing), does there exist a sequence of continuous functions $f_n:\mathbb{R}\to\mathbb{R}$ converging pointwise to $f$?      Here's the motivation for those questions: Let $(X,M)$ and $(Y,N)$ be measure spaces.   It's known that if $\mu$ is a measure on $(X,M)$ and $f:X\to Y$ is measure, then we have the pushforward measure $f_*\mu(A)=\mu(f^{-1}(A))$ on $(Y,N)$. What if we were to define a "pullback measure"? Given a function $f:X\to Y$ such that $f(M)\subseteq N$ (i.e. $f$ maps measurable sets to measurable sets) and a measure $\nu$ on $N$, the natural formula would be $f^*\nu(A)=\nu(f(A))$. So we ask if is there a good amount of functions which map measurable sets to measurable sets in $\mathbb{R}$, and monotone functions seem like good candidates (for strictly monotone, continuous functions, the result is valid and there are several answers on the web). If this were true, maybe we could use some convergence argument to solve the problem above.
Show that $K[x,xy,xy^2,\dots]$ is not Noetherian Here is the problem I am stuck on: Fix a field $K$ and consider the subring $A \leq K[x,y]$ generated by $K \cup \{x,xy,\dots,\}$. Show that $A$ is not Noetherian.  I figure that taking ideals $I_n = (x,xy,\dots,xy^n)$ should give an infinite strictly ascending chain, which would establish that $A$ is not Noetherian, but I cannot figure out how to show that each $I_n$ is a proper subset of $I_{n+1}$. Any help here? To EAV or not to EAV? So, somehow over the weekend, my 1GB USB flash drive has managed to become write-protected.  There is no switch on the stick, so I'm concluding that something has corrupted at some point.  Obviously I can't format it or remove.  The output from dmesg | tail is:  Buffer I/O error on device sdb1, logical block 328168 lost page write due to I/O error on sdb1 sd 73:0:0:0: [sdb] Unhandled sense code sd 73:0:0:0: [sdb] Result: hostbyte=DID_OK driverbyte=DRIVER_SENSE sd 73:0:0:0: [sdb] Sense Key : Data Protect [current] Info fld=0x0 sd 73:0:0:0: [sdb] Add. Sense: Write protected sd 73:0:0:0: [sdb] CDB: Write(10): 2a 00 00 05 08 00 00 00 01 00 Buffer I/O error on device sdb1, logical block 328200 lost page write due to I/O error on sdb1   I think the second and final lines are a giveaway to corruption, but I don't know how to interpret this output.  Could someone help to see if this output gives any indication?  I usually eject the drive before removing it, but I may have been lax once or twice.  The data isn't critical, it's just bugging me.  Any suggestions on how to fix it (if the drive even can be) would be greatly appreciated. Differences between LuaTeX, ConTeXt and XeTeX Arrows linking table cells I'm currently trying to add arrows in my synthetic division example to show people clearly how synthetic division works. I'm working on Overleaf.   The code for my synthetic division right now is:  \begin{tabular}{c |r r r r r}     &amp;1&amp; 4 &amp; -1&amp;16&amp;-20\\   1&amp;&amp; 1 &amp; 5 &amp; 4&amp;20\\   \hline    &amp;1 &amp;5 &amp; 4 &amp; 20 &amp;\rule{3pt}{16pt}\;0\\   \end{tabular}   How would I add arrows to this? For instance, a down arrow going from the first 1 to 1, then an upwards arrow pointing right from 1 to 1, then an arrow pointing down from 1 to 5, etc. Basically I would like to add arrows that look like this: How to secure my laptop so that hacking by physical access is not possible? Usually a probability distribution over discrete variables is described using a probability mass function (PMF):  When working with continuous random variables, we describe probability distributions using a probability density function (PDF) rather than a probability mass function. -- Deep Learning by Goodfellow, Bengio,  and Courville  However,  is using PDF to describe the probability distribution over discrete variables:    Is this a mistake? or it does not much matter? Is it possible to have several different views of one scene and set them as named "presets"? In-Page Editor Line Distortion The SFDC In-page editor has always had distortion for me, and I'm just wondering if it's just a lousy browser plugin they're using or if someone has found a good browser / operating system combination that stops the obnoxious distortion.   Obviously Salesforce doesn't have much interest in fixing it: as it's been this way on multiple operating systems and browsers I've used since I've used Salesforce.  However, my frustration with their other developer tools has lead me to use it occasionally and it's always the same: Consider the group $\mathbb{Q}$ under addition of rational numbers. If $H$ is a subgroup of $\mathbb{Q}$ with finite index, then $H = \mathbb{Q}$.  I just saw this on our exam earlier and was stumped on how to show this. Any ideas? When to use LinkedList over ArrayList in Java? I've always been one to simply use:  List&lt;String&gt; names = new ArrayList&lt;&gt;();   I use the interface as the type name for portability, so that when I ask questions such as these I can rework my code.    When should  be used over  and vice-versa? Not booting from USB or CD (SYSLINUX Message) I am trying to install linux on my laptop, a Toshiba Satellite C6550-S5200. I did it once but something happened so I removed it then I had to destroy all data on hard drive so now I have nothing on it. Well I got a iso file burned to a CD and to a flash drive. With the flash drive I get.   SYSLINUX 4.06 EDD 4.06-pre7 Copyright (C) 1994-2012 H. Peter Anvin et al  With the CD it will start booting it but somewhere loading it up, the dots turn all orange and stay that way and my CD drive turns quiet.  Oh and some more info the images work because I tried loading them up on another pc and it worked just fine.  I manage to get the CD to boot I just had to let me pc boot up first then insert the CD and have it boot the CD then. Once I get done installing ubuntu it works fine but I have to leave the PC on 24/7 for if I turn it off the PC will freeze 5-10 seconds after booting back up no matter how I install it. Can I get SSD rw performance while keeing data security if I combine SSD & HDD in btrfs RAID1 I have 2 storage devices; classical slow HDD (750GB, /dev/sda) and faster SSD (128GB, /dev/sdb). Currently I have installed Ubuntu &amp; Mint on same btrfs partition on SSD (/dev/sdb5). My btrfs pool consists of /dev/sdb5.  What I want to achieve is data replication on HDD while keeping SSD performance: Something like this:  For read case it would be ideal for procedure to go like this:   read from SSD check checksum if it verifies.. awesome, give me the data if not, fetch data from HDD and perform error correction through duplicate data   I'd manually run (some tool I'm not sure which one... btrfs check --repair perhaps?) every month or so to check HDD checksums and correct them via SSD data against bitrot.  For write case:   write to SSD sync to HDD when write requirements are low, thus not slowing down the system.   Is this possible and how would I do it?  Bottom line is, I like btrfs and would like it to duplicate data across HDD &amp; SSD drive automatically while keeping SSD excellent performance.   Using SSD as write cache is not an option since it's missing data duplication and security. round() doesn't seem to be rounding properly Prerequisites for classical mechanics by Susskind So I am an undergraduate in Electrical Engineering. We had a course on Physics in our freshman year which is equivalent to Classical Mechanics I as taught in MIT. I am interested in studying advanced classical mechanics (which includes Lagrangian formulation and other stuff) as equivalent to Classical Mechanics lecture by Susskind. I wanted to know are they any prerequisites which I should know about before taking this course.   In general what are the prereqs for this course? I am considering buying a Dell XPS 15 laptop. The laptop has a Nvidia 420m card which should work with Linux but I keep reading that the Nvidia Optimus technology isn't supported on Linux. I am not really interested in switching from Nvidia to Intel to save power but need to know that the Nvidia card will in fact work if I installed Ubuntu.  If anyone has experience using a Nvidia card with Optimus technology or even better the exact laptop in question (Dell XPS 15 with 1GB NVIDIA® GeForce®  GT 420M) it would be great.  A major problem holding people back from adopting Linux is this sort of hardware issue. I am a long term Ubuntu user and supporter who can't afford to make a mistake with a purchase like this. I don't want to spend £500+ to find I have no graphics acceleration because Windows 7 is not an option. How do you reverse a string in place in C or C++? How do you reverse a string in C or C++ without requiring a separate buffer to hold the reversed string? How to change file extension at runtime in Java $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ . Get top results for each group (in Oracle)