Let $f$ be a Lebesgue measurable function defined on a set $E\subset \mathbb{R}^d$. Let $\{x\in E\mid f(x)\neq 0\} $ be the support of the function on $E$.  Stein and Shakarchi claim without proof that the support is measurable. But why is this true? It's not just due to the fact that the support is a subset of a measurable set.
Could someone please check if I'm saying something that make sense? I am trying to check the following:     If $f$ is a measurable function, then $\text{supp}(f)$ is measurable set?   Is it simply because, if we consider $E_1 = (0,\infty)$ and $E_2 = (-\infty,0)$, since they are open, they are measurable. So their union $\mathbb{R}-\{0\}$ is measurable as well. Since $f$ is measurable, then the preimage $f^{-1}(\mathbb{R}-\{0\})$ is measurable. But the preimage is just the support?
The first given name of an author or editor in my bibliography should be written out, but all further first names should be abbreviated (initials with a period -- and space if there are multiple first names/initials, see below). I am using style=authoryear with BibLaTeX (v. 2.7) and Biber (v. 1.7).  My bibliography looks like this:     Doe, John Arthur Kyle (2013). The secret life of John Doe.   It should look like this:     Doe, John A. K. (2013). The secret life of John Doe.   Minimal example:  example.tex  \documentclass{scrartcl}  \usepackage[ngerman]{babel} \usepackage[babel,german=quotes]{csquotes} \usepackage[backend=biber,style=authoryear]{biblatex} \bibliography{bibliography}  \begin{document}  \cite[123]{Doe} \printbibliography  \end{document}   bibliography.bib  @book{Doe,     title = {The secret life of John Doe},     author = {Doe, John Arthur Kyle},     year = {2013} }   (Worked with said versions of BibLaTeX and Biber, Texmaker (v. 4.0.3) and MiKTeX (v. 2.9.4813).)  I could not find a solution in biblatex.def and firstinits=true abbreviates all first names, including the first one. Quite a few of the questions given on this site mention a photon in vacuum having a rest frame such as it having a zero mass in its rest frame. I find this contradictory since photons must travel at the speed of light in all frames according to special relativity.  Does a photon in vacuum have a rest frame? Prove no solution to $x^2\equiv3\bmod5$ Show that there is no solution to the congruence $x^2\equiv3\bmod5$.   I'm not sure if the fact that the only numbers divisible by $5$ are those ending in $0$ or $5$ will help&hellip; Sort array of objects by string property value Which media player has seek preview with thumbnails? I was looking for a good video player to replace VLC, something that is as good as vlc but has seek preview. I am tired of fast forwarding in vlc  EDIT: I meant seek preview with thumbnails What is the physical explanation for energy transport in simple electrical circuits? Keeping it simple, what is the physics approach to a current in a wire connected to a battery? How does the electric field behave?  I've read somewhere that, contrary to the common conception, electrons don't really flow as water in a hose so, what is really going on there? How is energy transported at the speed of light if electrons don't flow like very fast water? Why are floating point numbers inaccurate? Can I use the Google font 'Lato' for a logo design? I am creating a logo and would like to use the Google Font 'Lato', can I use this font in my logo design or are there copyright/limitations I need to worry about? I can't find anything online other than that it is an open source font that can be used in anyway you want (maybe that's my answer but I just want to make sure). Also, is it a bad idea to use a web-based font for a logo? Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$ Determine file type of an image I'm downloading some images from a service that doesn't always include a content-type and doesn't provide an extension for the file I'm downloading (ugh, don't ask).  What's the best way to determine the image format in .NET?  The application that is reading these downloaded images needs to have a proper file extension or all hell breaks loose. As my question says,   $$\frac 1{1.2.3}+\frac 1{2.3.4}+\frac 1{3.4.5}+...$$  As far as $\frac 1{1.2}+\frac 1{2.3}+\frac 1{3.4}+...$ is concerned, we can write it as $\frac 1{n-1} - \frac 1{n}$ What can we do here? Where has the trailing newline char gone from my command substitution? I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as well? So Black Holes Actually Merge! In 1/5th of a Second - How? I've read a lot of conflicting answers in these forums. However, today saw the awesome announcement of gravitational waves. Two black holes merged:   Not only that, they merged FAST. In 1/5th of a second revolving around each other 250 times a second. The entire event was quicker than a heartbeat. Moreover, we observed this happening as distant outsiders. So now we can say for sure:   Objects approaching the event horizon DO NOT appear to slow down Black holes CAN merge in a finite (and quick) amount of time And all this is wrt a frame of reference far, far away   To quote the NYTimes article:  One of them was 36 times as massive as the sun, the other 29. As they approached the end, at half the speed of light, they were circling each other 250 times a second.  And then the ringing stopped as the two holes coalesced into a single black hole, a trapdoor in space with the equivalent mass of 62 suns. All in a fifth of a second, Earth time.  However, everything I've read so far has let me to believe that an outside observer should never be able to measure the collision happening in a finite time. So what exactly is happening here? I must have read at lest 5 different versions of this so far everywhere in these forums over the past several years. If I have Optimize iPhone Storage enabled on my iPhone and my phone storage is completely full, I understand that iOS will only store a low-res thumbnail of many of my photos on the device. (The full high-res version of the image will only exist in iCloud and on devices with Download and Keep Originals enabled.)  But when only thumbnails exist on the device, how does that work in relation to other photo apps on iOS? For example, do third-party cloud-based photo apps (like Google Photos, Dropbox's Carousel, or Amazon Prime Photos) have access to the full high-res images from the iCloud Photo Library? Or will they only see the low-res thumbnails in some cases? The latex file coding is not there but the pdf file is there I wrote all my coding in the .tex file. The next day when I opened it the coding is all gone and the pdf file is opening with it and is displayed as an output. How do I bring back my coding? What are the various physical mechanisms for energy transfer to the photon during blackbody emission? By conservation of energy, the solid is left in a lower energy state following emission of a photon.  Clearly absorption and emission balance at thermal equilibrium, however, thermodynamic equilibrium is a statement of the mean behaviour of the system, not a statement that the internal energy is constant on arbitrarily short timescales.  The energy has to come from somewhere during emission, and go somewhere during absorption.  Energy in a solid can be stored as kinetic and potential energy of electrons and nuclei, either individually or in collective modes such as phonons and plasmons.  In thermal equilibrium energy will be stored more or less in various forms depending on the temperature and material.  However, even if most of the thermal energy in a particular solid at temperature $T$ is stored in the form of phonons, it could be that phonons primarily interact with light indirectly via electrons, e.g. a phonon excites an electron in a phonon-electron interaction, which can interact with light via the EM field.  Given that light is an EM field, it makes sense to me that it is emitted and absorbed by charged particles.  The electron-photon interaction is probably dominant for visible and ultraviolet light, given that metals are opaque, while semiconductors and insulators are transparent to (visible and UV) light with energy lower than their bandgap. However, once you get into energies in the IR and below, or X-rays and above, other mechanisms apparently take over.  For example, on the high-energy end of the spectrum I've heard that gamma rays can interact directly with nuclear degrees of freedom, which is reasonable considering that gamma rays are emitted during a lot of nuclear reactions.  A review of  might give clues to important light-matter interactions over a broad range of wavelengths.  Whether all of the these processes are involved in blackbody emission is a somewhat different question.  What physical processes mediate energy transfer during blackbody emission, and in which energy ranges are the various processes dominant? How to draw a shaded sphere? Let me say first that I am fairly new here (about 3 weeks) and absolutely love Stack Overflow. At first I let some of the negative attitudes get to me a little, left a few snarky comments, but that phase has passed. I see the tremendous value for this site in the long run and sincerely do want to help make it a better place.  I find myself going for  the downvote very frequently -- currently about 35% of my votes are downvotes. I even make an extra effort to upvote good content to offset this a bit, so the number should probably be more like 60%. I always try to leave constructive comments and try to help out the confused ones who are really trying.  I truly can't tell if it is just me, or if these folks deserve it. I don't know if you guys noticed, but there are a lot of really bad questions being posted, and a lot of people trying to pass off things like "use jQuery", "use Firebug", and "use Google" as good answers (and getting upvotes!). I even saw someone copy/paste my "What does Firebug say?" comment as an answer and get 3 votes within moments. Sometimes This answer is unclear or not useful just doesn't do it justice.  I feel like some of the people just drive me nuts, but then again I feel this way IRL :) Am I being too harsh, or is it really that bad on SO? Has there recently been a surge of new users, and the quality of content is degrading, or was it like this in the "good old days"? Have I perhaps misunderstood the expected behavior on Stack Overflow -- should I just ignore these crappy questions and answers rather than downvote them? Compute $\frac{1}{e}\sum\limits_{n=0}^{\infty}\frac{n^{k}}{n!}$ for $k=0, 1, 2 ... $ Using some matlab (I know it's cheating) I found that: $$k=0 =&gt; Result=1$$ $$k=1 =&gt; Result=1$$ $$k=2 =&gt; Result=2$$ $$k=3 =&gt; Result=5$$ $$k=4 =&gt; Result=15$$ $$k=5 =&gt; Result=52$$ $$k=6 =&gt; Result=203$$  This sequence is puzzling...