Here is the question:  Prove that for every $\epsilon&gt;0$ and every set $A\in\mathcal{B}(\mathbb{R}^{n})$ there is a compact set $K\subset A$ such that $P(A\setminus K)\leq\epsilon$.   --I have previously shown that there is a closed set $F$ and an open set $G$ such that $F\subset A\subset G$ and $P(G\setminus F)\leq\epsilon$. For the current problem, I think that we can find $K$ as an approximation of the set $F$, such that $K=F_{N}=F\cap[-N,N]^{n}$, which is compact. I think I am just caught up in the set difference algebra, i.e.,$$P(A\setminus K)=P(A\setminus (F\cap[-N,N]^{n}))=P(A\cap(F^{c}\cup[-N,N]^{c})^{c})=\cdots$$  --Any help on how to finish this is appreciated!
This is an exercise in real analysis:     Let $E\subset{\Bbb R}^d$ be Lebesgue measurable. Show that    $$ m(E)=\sup\{m(K):K\subset E, K \text{compact}\}. $$   When $E$ is bounded, this can be done by the following proposition:       $E\subset{\Bbb R}^d$ is Lebesgue measurable if and only if for every $\varepsilon&gt;0$, one can find a closed set $F$ contained in $E$ with $m^*(E\setminus F)\leq\varepsilon$.    How can I deal with the case that $E$ is unbounded?
What's the appropriate reaction to an answer that doesn't attribute the source it's copied from? Do superheroes/villains like Quicksilver or Flash with super speed get tired due to running? I mean a average human can run at his full speed for 15 to 20 mins before getting tired. Trained athletes can go on running for much more time without tiring out.  Do Superheroes like Quicksilver or Flash ever get tired due to excessive running? A world where nothing can be hidden. No lies, always truth In a Bollywood movie Pk, a man comes from a planet where lying is not possible as all the thoughts/knowledge is transferable via human contact. Due to which they do not have to lie about anything or hide anything from each other.  Say, such a society plausible with technology, if we decide to abandon the privacy as we know today. All the recorded data (written, spoken, recorded videos etc.) and thought (via neuralink) is accessible to everyone, everywhere.  Basic idea behind the question is, people worry about data being accumulated at one place (entity or government) and being used against them or to deceive them. If everything is accessible/known to everyone of about anyone and anything, what does privacy mean in this case? Showing that $\gamma = -\int_0^{\infty} e^{-t} \log t \,dt$, where $\gamma$ is the Euler-Mascheroni constant. I'm trying to show that  $$\lim_{n \to \infty} \left[\sum_{k=1}^{n} \frac{1}{k} - \log n\right] = -\int_0^{\infty} e^{-t} \log t \,dt.$$  In other words, I'm trying to show that the above definitions of the Euler-Mascheroni constant $\gamma$ are equivalent.  In another post here (which I can't seem to find now) someone noted that  $$\int_0^{\infty} e^{-t} \log t \,dt = \left.\frac{d}{dx} \int_0^{\infty} t^x e^{-t} \,dt \right|_{x=0} = \Gamma&#39;(1) = \psi(1),$$  where $\psi$ is the digamma function.  This may be a good place to start on the right-hand side.  For the left-hand side I was tempted to represent the terms with integrals.  It is not hard to show that  $$\sum_{k=1}^{n} \frac{1}{k} = \int_0^1 \frac{1-x^n}{1-x} \,dx,$$  but I'm not sure this gets us anywhere.  Any help would be greatly appreciated. Rotating symbols per attribute value in QGIS? I am trying to apply a rule to rotate symbols based on the value of an attribute.  I tried to copy , which is essentially my question, however it does not work for me.   I have a layer with two attributes, the slope and aspect.  I want to colour the symbols by the slope (this part works) and rotate by the aspect.  The steps I take are:   Layer Properties -> Style -> Graduated -> Symbol -> Change Select Simple Marker (arrow) and press the button to the right of angle -> field type (double) -> Aspect-sel    Applying these commands changes the symbol to an arrow, and I can easily apply the colour gradient, but the symbols do not rotate. The attribute Aspect-sel varies from 2.87 to 334.23.    ETA: I was using Version 2.8.3 - on suggestion I updated to 2.14.10, which solved the problem. Why was the alien threat in The Next Generation episode "Conspiracy" never followed up? In the Star Trek The Next Generation episode  Picard and crew are drawn into an alien conspiracy to take over the Federation. This ends with Picard and Riker destroying an alien "mother" beast, but not before it manages to send a signal to an uncharted sector of space. The feeling at the end of the episode is that this will be a threat that the intrepid crew of the Enterprise would be facing again at some point, but this never happens.   Why was the implied threat at the end of this episode never brought up again on the show? Isle vs. Island Application always show in the third position in Ubuntu 16.04 After I update my ubutu to 16.04, I got a problem with Unity launcher. Whenever I open an application which is not listed in launcher, it will take the third position instead of last position. Has anyone the same problem? Interface vs Abstract Class (general OO) Circular determinant problem I'm stuck in this question: How calculate this determinant ? $$\Delta=\left|\begin{array}{cccccc} 1&amp;2&amp;3&amp;\cdots&amp;\cdots&amp;n\\ n&amp;1&amp;2&amp;\cdots&amp;\cdots&amp; n-1\\ n-1&amp;n&amp;1&amp;\cdots&amp;\cdots&amp;n-2\\ \vdots &amp;\ddots &amp; \ddots&amp;\ddots&amp;&amp;\vdots\\ \vdots &amp;\ddots &amp; \ddots&amp;\ddots&amp;\ddots&amp;\vdots\\ 2&amp;3&amp;4&amp;\cdots&amp;\cdots&amp;1 \end{array}\right|$$ Thanks a lot. gluten free hot cross buns How do I move a line in a text file up or down by one line? Put a where clause around a stored procedure I cannot edit What does f-stop mean? Zero morphism does not depend on zero object I want to prove that zero morphism does not depend on zero object. Let a zero object $0 \in \mathcal{C}$.  I supposed that exists another zero object $0' \in \mathcal{C}$, and I considered $0: A \to B$ and also $$A \to 0 \to 0' \to B$$ But I don't know to prove that zero morphism $0: A \to B$ not does not depend on zero object. Can anybody give me a suggestion? Turn off paragraph indention for a single line Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ . I'm looking for a simple proof that up to isomorphism every group of order 2p (p prime) is either $\mathbb{Z}_{2p}$ or $D_{p}$ (The Dihedral group of order 2p).  I should note that by simple I mean short and elegant and not necessarily elementary. So feel free to use tools like Sylow Theorems, Cauchy Theorem and similar stuff.  Thanks a lot! What is a "good fit" Brier score and Harrell's C Index Launch an app from within another (iPhone) Is it possible to launch any arbitrary iPhone application from within another app?, For example in my application if I want the user to push a button and launch right into the Phone app (close the current app, open the Phone app).  would this be possible? I know this can be done for making phone calls with the tel URL link, but I want to instead just have the Phone app launch without dialing any specific number. Why is the ampere a base unit and not the coulomb? I always thought of current as the time derivative of charge, $\frac{dq}{dt}$. However, I found out recently that it is the ampere that is the base unit and not the coulomb. Why is this? It seems to me that charge can exist without current, but current cannot exist without charge. So the logical choice for a base unit would be the coulomb. Right?