We know that Lebesgue Outer Measure is countably sub-additive. It is not even finitely additive. I am trying to find two sets of real numbers which are disjoint and yet $\mu ^*(A\cup B) = \mu^*(A) +\mu^*(B)$ doesn't hold.
Let $\{E_i\}_{i=1}^n$ be finitely many disjoint sets of real numbers (not necessarily Lebesgue measurable) and $E$ be the union of all these sets. Is it always true that  $$ m^\star (E)=\sum_{i=1}^N m^\star(E_n) $$ where $m^\star$ denotes the Lebesgue outer measure? If not, please give a counterexample. The Vitali set is a counterexample in the countable case, but I am not sure whether it is false in finite case.
Why not use development provisioning instead of ad hoc? Gauss's Lemma for polynomials claims that a non-constant polynomial in $\mathbb{Z}[X]$ is irreducible in $\mathbb{Z}[X]$ if and only if it is both irreducible in $\mathbb{Q}[X]$ and primitive in $\mathbb{Z}[X]$.   I wonder if this holds for multivariable case.      Is it true that a non-constant polynomial in $\mathbb{Z}[X_1,\dots,X_n]$ is irreducible in $\mathbb{Z}[X_1,\dots,X_n]$ if and only if it is both irreducible in $\mathbb{Q}[X_1,\dots,X_n]$ and primitive in $\mathbb{Z}[X_1,\dots,X_n]$?    Thank you for your help. A few years ago I saw a scifi movie about a group of travelers stranded at what I believe might have been a bus station (I don't remember that detail with any certainty).  There were 2 children (a boy and a girl) who were running around and everyone thought they belonged to someone else there.  As it turned out, the children were aliens that reverted to their true form and began hunting down the travelers.  I've been searching all over the internet but can't come up with the name of the movie.  Does anyone know the name of it? Clarifications about Gstreamer I'm a bit confused about what Gstreamer does.   Does it come installed by default with Ubuntu ?  If so which version  is the default for 14.04?  Does it provide hardware accelerated video playback (decoding) ?   If so does it work with VA-API and VDPAU or is it an alternative to those?    Thanks for your help How can I tell whether beetroot is still edible? I have some raw beetroot I bought about a week ago, having stored them at room temperature since. It's the first time I bought it raw, and I'm unsure whether it's still edible.   It is already soft, and when I cut it, it smells... Well, different than I expected. It's not a rotten smell as such, but it's nothing like the sweet beet-rooty smell I know from the pre-packaged, pre-boiled supermarket variety.  How can I tell whether it's still safe to eat? Where does the extra kinetic energy of the rocket come from? Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding $100\ \text{m/s}$ while initially travelling at $1000\ \text{m/s}$ will add a great deal more energy to the ship than adding $100 \ \text{m/s}$ while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from. How can I implement a download rate limited in Java? How do they film continuous blood pooling? Why do the measurements of this object seem erroneous? Torsion elements of a group aren't necessarily a subgroup In a question from an exam in an undergraduate group theory course, we were asked to prove or disprove that the set of all Torsion elements of a group is necessarily a subgroup.  I knew that the set of Torsion elements is closed under the inverse operation, but was later told that it is not closed under multiplication, therefore disproving the claim. However, I couldn't find any examples of a group $G$ and two elements $a,b$ such that both $a$ and $b$ have a finite order, but $ab$ doesn't. I know that for this to happen $G$ must be an infinite and non-Abelian group, but still couldn't find a valid example.   What are some examples of groups/elements fulfilling the aforementioned property? Disable OS X startup sound How can I permanently disable the startup sound (or chime) in Yosemite?  (I am using a late-2014 13" macbook pro)  I already tried several options found here and there :    installed StartNinja.app => did not work installed Auto-Mute.app  => did not work tried to install the old StartupSound prefPane => did not work tried the command "sudo nvram SystemAudioVolume=%80" => volume does not change (I also tried other values than 80, such as 00, 01, or nothing)   NOTE: I know that it is possible to mute the next startup sound by turning the volume down, but this is not a permanent solution. As per title, I wish to know if I can, after a successful 32bit setup and consequent software installations, update to a 64bit version.  I know how to partition (actually one of the solutions is to set /, /etc, /home, /var/www, and /opt as separate partitions) and I know that a clean install is way better than a dirty one, yet I would like to know if/how it's possible to do that. Can this statistic be shown not to be sufficient for $\theta$? This problem comes from , who do not rigorously demonstrate (in their solution key) that the statistic is not sufficient.  Let $X_1,\dots,X_n$ be a random sample from a population with PDF $f(x|\theta)=\theta x^{\theta-1} \cdot 1_{\{x\in(0,1)\}}$ for $\theta&gt;0$. Show that $\sum_i X_i$ is not sufficient for $\theta$.  If you write out the PDF $p(\vec{x}|\theta)$ of the random sample, it is clear by the factorization theorem that $\prod_i X_i$ or $\sum_i \log(X_i)$ are sufficient for $\theta$; the PDF $p$ also suggests $\sum_i X_i$ is not sufficient for $\theta$. However, to rigorously demonstrate this, we need to analyze $p(\vec{x},\theta)/q(T(\vec{x},\theta)$, where $p$ is the distribution of sample $\vec{x}$, $q$ is the distribution of the statistic $T(\vec{X})=\sum_i X_i$. But finding the distribution of $T(X)=\sum_i X_i$ seems impractical. I have observed that $f$ is the PDF of a Beta($\theta$,1) distribution, but checking online, the distribution of the sum of Beta random variables doesn't appear to have a closed form. Are there any alternative routes (e.g., showing there is no factorization involving $T(\vec{X})$)? Did C&amp;B leave out a full explanation that $\sum_i X_i$ is not sufficient because none actually exist? Let $a,b,c\in\mathbb{R}^*_+$, $abc=1$.  How can i show that $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$ ?  I got $\left(ab-b+1\right)\left(bc-c+1\right)\left(ca-a+1\right)\leq abc=1$, but i can't go any further ...  If i expand it :   $2-ab-\frac{b}a-c-ac-\frac{a}c-b+2a+\frac{2}a-bc-\frac{c}b-a+2b+\frac{2}b+2c+\frac{2}c-4$  Which gives using $c=\frac{1}{ab}$ :   $5ab-\frac{b}a-\frac{1}{ab}-a^2b+\frac{1}a+b+\frac{1}b-2$ Length of a JavaScript object I have a JavaScript object. Is there a built-in or accepted best practice way to get the length of this object? const myObject = new Object(); myObject[&quot;firstname&quot;] = &quot;Gareth&quot;; myObject[&quot;lastname&quot;] = &quot;Simpson&quot;; myObject[&quot;age&quot;] = 21; What makes the number of neutrons the number of proton similar? Can you explain Parzen window (kernel) density estimation in layman's terms? Inverse of $f(x)=\sin(x)+x$ What is the inverse of   $$f(x)=\sin(x)+x.$$  I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet.  What about  $$f(x)=\sin(a \cdot x)+x$$  where $a$ is a known real constant.  Thank you for taking the time to read this question! Sorry if this has been asked before... How to disable Django query cache? I'm looking for the name of a sci-fi series, intended for children. Robots have control over the humans. There is a squad of humans who attack them. Every member in this squad has a badge on their suit, and when they touch this badge the suit becomes armed. This series was not cartoon series, it was performed by real people. The main robot character is very similar in style to Darth Vader in Star Wars. The series aired in the 1990s. I am sorry but I can't recall many details.