How can I construct a set $E$ such that $E$ is dense in $[0,1]\times [0,1]$, and the intersection of $E$ and any line parallel to the axes has at most one point?   I am finding this set in order to construct a counterexample to show that the Tonelli's Theorem is not true for Riemann integral because we can define a function $$f(x,y)=\begin{cases} 1, &amp;(x,y)\in E,\\ 0, &amp;(x,y)\notin E, \end{cases}$$ then $f\notin R([0,1]\times [0,1])$, but we have the following: $$ \int_0^1dy\int_0^1f(x,y) dx=0=\int_0^1dx\int_0^1f(x,y)dy.$$  Apparently, I also want the measure of $E$ to be $0$.
Given a subset $A$ of $\mathbb{R}^{n}$, a point $x \in \mathbb{R}^{n}$ is said to be in the boundary of A if and only if for every open rectangle $B\subseteq\mathbb{R}^{n}$ with $x\in B$, $B$ contains both a point of $A$ and a point of $\mathbb{R}^{n}\setminus A$.  My question is from Spivak's Calculus on Manifolds: Construct a set $A \subseteq [0,1]\times [0,1]$ such that $A$ contains at most one point on each horizontal and vertical line but has boundary equal to $[0,1]\times[0,1]$.
Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable? Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)&gt;c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable?  There are a number of equivalent definitions for the measurability of a function and the most obvious one would be to show that $\{x\in E : f(x)&gt;c\}$ is measurable for all $c\in\mathbb{R}$.  Thus my strategy has been to consider an arbitrary irrational $y$ and use the density of the rationals in the reals to show that there exists some open set $O$ such that $m^*(O-\{x\in E : f(x)&gt;y\})&lt; \varepsilon$.  I would do this by choosing some rational $q&lt;y$, close enough to $y$, such that the set of values in $E$ which get sent under $f$ to values strictly greater than $q$ and less than or equal to $y$ is small enough.  But that doesn't in general seem possible since I could keep sending intervals of some fixed positive length to smaller and smaller regions just beneath $y$.  However conversely, such a function would not be enough for a counter example, since finding such a function only rules out this strategy, but doesn't necessarily entail non-measurability.  Is there a better definition of the measurability of a set I should use? or does the statement actually not imply measurability?  Thanks. Is it possible to remove a terminal command? For example, if there was a command call lookup (not an alias), how would I go about removing it? How can I install a .bin file? Serial voteup someone you learn from I am a girl in my early twenties traveling to USA from India and in my visa application I have provided the contact address for a relative in San Francisco, and their invitation letter. I would like to visit LA first. Will I face problems during immigration if I land in LA (with an immediate on wards ticket to SF)? I dont plan to get on the flight from LA to SFO.  Initially I was planning to fly to SFO from India and then fly to LA immediately after completing immigration at SFO airport. Orthogonality and linear independence [Theorem] Let $V$ be an inner product space, and let $S$ be an orthogonal subset of $V$ consisting of nonzero vectors. Then $S$ is linearly independent.     Also, orthogonal set and linearly independent set both generate the same subspace.  (Is that right?)    Then orthogonal $\rightarrow$ linearly independent but orthogonal $\nleftarrow$ linearly independent is that right?    One more question. For T/F, Every orthogonal set is linearly independent (F) Every orthonormal set is linearly independent (T) Why? How can I get the source code of a Python function? Suppose I have a Python function as defined below:  def foo(arg1,arg2):     #do something with args     a = arg1 + arg2     return a   I can get the name of the function using foo.func_name. How can I programmatically get its source code, as I typed above? So I did a fresh install of Ubuntu 20.04 on a computer with the new RTL8125B Realtek network card and I cannot get the network to work. I was very suprised since this is supposed to be the new feature of the 5.4 kernel (I have 5.4.0-26-generic installed). Does anyone have any idea on how to get it working? Thank you. $ lspci -knn | grep Eth -A3 06:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8125 2.5GbE Controller [10ec:8125] (rev 04) Subsystem: ASUSTeK Computer Inc. RTL8125 2.5GbE Controller [1043:87d7] Kernel modules: r8169 07:00.0 VGA compatible controller [0300]: Advanced Micro Devices, Inc. [AMD/ATI] Ellesmere [Radeon RX 470/480/570/570X/580/580X/590] [1002:67df] (rev e7)  $ dmesg | grep -i r8169 [    0.911236] r8169 0000:06:00.0: unknown chip XID 641 Finding $\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2} dx$ Calculate  $$\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2}dx$$   I have tried to put $\displaystyle x=\frac{1}{t}$ and $\displaystyle dx=-\frac{1}{t^2}dt$  $$ \int^{\infty}_{0}\frac{t^2\ln^2(t)}{(t^2-1)^2}dt$$  $$\frac{1}{2}\int^{\infty}_{0}t\ln^2(t)\frac{2t}{(t^2-1)^2}dt$$  $$ \frac{1}{2}\bigg[-t\ln^2(t)\frac{1}{t^2-1}+\int^{\infty}_{0}\frac{\ln^2(t)}{t^2-1}+2\int^{\infty}_{0}\frac{\ln(t)}{t^2-1}dt\bigg]$$  How can I solve it? Is there a vendor for animal parts? I'm looking to upgrade all my pouches, but some animals are hard to find, or the part needed drops rarely.  Is there a merchant somewhere that sells them outright? I have shards to spare. Please add a system to allow file uploads attached to questions and answers If I use  \usepackage[colorlinks]{hyperref}   I get colored links for \cite (not good for printing or even viewing) but table of content looks fine.  If I use  \usepackage[colorlinks=false]{hyperref}   I get \cite  links in black with a colored frame (this looks very good) but the table of content looks strange with extra long boxes.  What are my other options? Can you have two styles in different parts of a document?   How do you make the page numbers in TOC to be the links instead of the section headers? Why is my query suddenly slower than it was yesterday? [Salutations]  (check one)  [ ] Well trained professional, [ ] Casual reader, [ ] Hapless wanderer,   I have a (check all that apply)  [ ] query [ ] stored procedure [ ] database thing maybe     that was running fine (if applicable)   [ ] yesterday [ ] in recent memory [ ] at some point    but is suddenly slower now.  I've already checked to make sure it's not being blocked, and that it's not the victim of some long running maintenance task, report, or other out of band process.  What is the problem, what should I do, and what information can I provide to get some help?  [*Insert appropriate closing remarks*] "They're going gorilla watching" - is there a verb at the end of the sentence? In this sentence:     They're going gorilla watching.   what's the function of the verb watching?   Is it a verb? a noun? ... maybe gorilla watching means "mirador de gorilas", in spanish?  In English, I have not seen a verb at the end of a sentece. How do I check if a package is installed on my server? I recently installed Ubuntu Server edition 13.10 (Saucy Salamander). But I have run into a bit of problem. At the end of the installation, it asks about several extra packages whether you want them or not, like OpenSSH server and virtual machine host. Here's what I'm looking at:  Now, my problem is that I can't remember if I have checked the &quot;virtual machine host&quot; checkbox. How do I check if my server has the corresponding packages? What packages come with the virtual machine host? Is "flu" the only disease usually accompanied with "the"? From what I understand, you don't usually use "the" with diseases / illnesses / disorders ("have Alzheimer's / diarrhea / claustrophobia / dyslexia, etc." not "have the..."), but it's perfectly normal to say "have the flu". Is the flu the only exception?   Edit (this is only a side part, and can be glossed over if you will, so it's not a duplicate imo): What's so significant about the flu that other conditions don't have, so that it's entitled to "the"? Does "apropos" take a preposition? How do you use this word, anyway? Which is more correct:     Apropos of your earlier comment, I decided to....   or      Apropos your earlier comment, I decided to...   Actually, apropos is so fancy a word, even I, a word maven if I do say so myself, am not entirely sure when to use it.  If I may add to my preposition question, are there any constructions where it would be appropriate to use something like with regard to and not apropos? I recently installed Ubuntu Server edition 13.10 (Saucy Salamander). But I have run into a bit of problem. At the end of the installation, it asks about several extra packages whether you want them or not, like OpenSSH server and virtual machine host. Here's what I'm looking at:  Now, my problem is that I can't remember if I have checked the &quot;virtual machine host&quot; checkbox. How do I check if my server has the corresponding packages? What packages come with the virtual machine host? python: restarting a loop i have:  for i in range(2,n):     if(something):        do something     else:        do something else        i = 2 **restart the loop   But that doesn't seem to work. Is there a way to restart that loop?   Thanks How to configure splash screen after plymouth I have configured my splash screen for plymouth - and it works perfectly. I've also configured the display manager (I'm using gdm). In between these two I get a very brief splash screen - a kind of swirl of multi-coloured ubuntu like colours. Where does this splash screen get set, and how do I change it?  Edit: Here is a video of my boot sequence, showing the screen in question: