Construct a measurable set $E\subset [0,1]$ such that for any   non-empty open sub-interval $I$ in $[0,1]$, both sets $E\cap I$ and $E^c\cap I$ have positive measure.      [Stein's Hint: For the first part, consider a Cantor-like set of positive measure, and add in each of the intervals that are omitted in the first step of its construction, another Cantor-like set. Continue this procedure indefinitely.]   So I tried using the hint to construct the desired set, but I must be misunderstanding something because I do not think the construction works.     My Thoughts:  I believe the basic idea is that we generate a Cantor-like set $C_1$ by removing repeatedly open intervals of some appropriate length at each stage of the construction starting from $[0,1]$.  Then as said in the hint, at the first step in the construction we removed centrally some open interval $I_1$ from $[0,1]$. Then we generate another Cantor-like set $C_2$ from $I_1$. During the first stage of $C_2$'s construction we removed $I_2$ from $I_1$. Repeat indefinitely...  My problem with this is that during the second stage of the construction of $C_1$, we removed $2$ open intervals, call one of them $\mathcal{I}$, from $[0,1]\setminus I_1$. If we take the union of all these Cantor-like sets, denote it as $E$, then wouldn't $E\cap \mathcal{I}=\emptyset$?  If true, then it seems we would have to apply this to all the open intervals that $C_1$ removes and generate a collection of Cantor-like sets from them. Each of which would need the same procedure done to them.   This doesn't seem correct to me and we may need to define $E$ as the countable union of countable unions of Cantor-like sets.    Any help thinking correctly about this problem would be much appreciated.  Note: It's proven (I've proved) that the Cantor-like sets Stein is talking about have positive measure (Exercise 4 in Stein).
This is from problem $8$, Chapter II of Rudin's Real and Complex Analysis.  The problem asks for a Borel set $M$ on $R$, such that for any interval $I$, $M \cap I$ has measure greater than $0$ and less than $m(I)$.  I was thinking of taking the Cantor approach: taking $R$ to be the union of $[a,b]$ with $a$ and $b$ rationals, and for each $[a,b]$ we construct Cantor sets inside it. During theconstruction of each Cantor set, in order to have positive measure on it, we need to take off smaller and smaller intervals from it, namely the proportion goes to $0$. As a result, these Cantor sets are extremely "dense" on their ends. If for an interval $I$ it intersects with the Cantor set on $[a,b]$ while $b-a&gt;&gt;m(I)$, we shall expect the measure of intersection to be rather close to $m(I)$ and then we lose control on these cases.  Is there any way to fix this or shall I consider other approaches?  Thank you
How to partition for Windows/Ubuntu dual boot on same SSD I have a 500GB SSD in which i would like to have both Windows and Ubuntu. My plan is to have something similar to this: Windows 8.1 and ubuntu 14.04 Uefi boot   an efi partition (created by windows installer) which will have Grub as bootloader 150GB ntfs for Windows 15GB ext4 for Ubuntu (root) 35GB ext4 for Ubuntu (home) ~8-16GB swap for Ubuntu (I have 16GB RAM) ~284-292GB ntfs for Data shared for both Windows and Ubuntu   My current setup is   200GB ntfs Windows 300GB ntfs for Data   I plan to shrink both the Windows and the data partiton in order to achieve my desired plan.  My questions:   Does the order of partitions matter? Does it matter if the ubuntu partitions are not at the end/beggining of the drive? Does the swap partition behaves better if its the last/first partition? Can I use the shared NTFS Data partition between windows and ubuntu without encountering problems? Do I have enough space for my ubuntu /home partition (thinking that I'll have access to the ntfs shared partition)?   Edit: Laptop, Asus N56jr, i7, 16Gb Ram, SSD samsung 850 evo 500gb  Windows 8.1 and Ubuntu 14.04 I want to enable gzip compression for acethehimalaya.com. Can anybody please tell me what is the best way of achieving compression in an Apache Webserver? Diagram of an artificial neural network How can something finite become infinite? How can the universe become infinite in spatial extent if it started as a singularity, wouldn't it take infinite time to expand into an infinite universe? How to deal with very long lines that do not contain spaces? How to catch creation of DOM elements and manipulate them with jQuery Load CSS/Javascript in frontend conditionally if block is used Assuming I have no infinite scroll or anything else going on in my theme: Is there a way to enqueue my custom Gutenberg block styles and scripts so they are only (lazy) loaded in the front-end when the block is really used? Something like is_active_widget, just for blocks?  If I enqueue them inside a enqueue_block_assets function they're added to any site.  On github, I found this  , but it sounds like it's more on bundling than conditional load, so I still hope they didn't leave that optimization opportunity out - otherwise any other site will soon be junked with 20 additional scripts from numerous Gutenberg block plugins people install and only use once on a page (think of large image gallery block scripts for example).  Thanks &amp; Regards! Why is the well ordering principle counter-intuitive? I read  that while 'The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians'. I don't understand why this is so. According to Wikipedia, 'the well-ordering principle states that every non-empty set of positive integers contains a least element'. This is quite a straightforward definition. Is it the way you derive the equivalence of the Axiom of Choice and well ordering principle that makes it counter-intuitive or the definition of the least element? In Quantum Mechanics we usually consider operator-valued functions: these are functions that take in real numbers and gives back operators on the Hilbert space of the quantum system.  There are several examples of these. One of them is when we work with the Heisenberg picture, where we need to consider functions $\alpha : \mathbb{R}\to \mathcal{L}(\mathcal{H})$ such that $\alpha(t)$ is the operator at time $t$.  Another example is when we deal with exponentiation of operators, like when building the time evolution operator:  $$U(t,t_0)=\exp\left(-i\frac{H}{\hbar}(t-t_0)\right),$$  Here, the $\exp$ is usually understood as being defined via the eigenvalues of $H$.  The point is, the idea of a function $\alpha : \mathbb{R}\to \mathcal{L}(\mathcal{H})$ appears quite often in Quantum Mechanics, and sometimes one needs to differentiate these. In practice we do it formally, using all the properties we would expect, but I'm curious about how one would properly define this.  If we were dealing with bounded operators, then we could use the operator norm, which is available for this kind of operator, and define the derivative as we can usually we do when there's some norm around.  The point is that in Quantum Mechanics most of the time operators are unbounded.  So in the general case, how can one define the derivative of one operator-valued function? Viruses in wine? Make the chat link one click away in the new top bar Does That's very sweet of you  in the US sound like That's very sweef you? the of is only speak f and link with swee I am aware of the die() command in PHP which exits a script early.  How can I do this in Python? Use of # a.k.a. read-macro Why is this trigger finding a null value despite the field being populated? Truncated 200GB table but disk space not released I have only 2GB left, so I need to remove this history table. This table now is empty but the database disk space not released. And the database file is 320GB. Is there difference in force when the height increases? Why is TV remote's infrared wave shown by a camera as purple light, why not red or something else? hotel occupancy counting problem The multiple-choice question:  A hotel has four vacant rooms. Each room can accommodate a maximum of four people. In how many different ways can six people be accommodated in the four rooms? (A) 4020 (B) 4068 (C) 4080 (D) 4096.  My attempt:  If there were no 'four person' constraint, the six people could be accommodated in $4^6=4096$ ways. however, this number includes combinations with 5 or 6 people in one room, which should be excluded.  If one room has 5 occupants, one other room has 1 occupant. The number of '1,5' combinations is ${{4} \choose {2}}$=12. If one room has 6 occupants, the other three rooms have no occupants, and there are four such combinations.  I believe the answer should be $4096-12-4=4080$, but am not confident. I'd be grateful for any comments on my attempt. Is an expanding map on a compact metric space continuous? I got inspired by this question  to think about the following problem: Suppose you have a compact metric space $(X,d)$ and an expanding map $T:X\rightarrow X$, i.e. $d(Tx,Ty)\geq d(x,y)$ for every $x,y\in X$. Is the map $T$ then continuous? Without assuming continuity we already know that $T^n(X)$ must be dense in $X$ for any $n\geq 1$, since given $z\in X$ the orbit of $z$ must accumulate upon $z$. But showing continuity has escaped me so far. If there are counter-examples using the Axiom of Choice that also interests me.