I am working on this problem$^{(1)}$ on measurable function like this:     Show that $f$ is measurable if $(X, \mathcal A)$ is measurable space, $f$ is real-value function and $\{x \mid f(x) &gt; r \} \in \mathcal A$ for each rational number $r.$   Earlier on the text gives this definition$^{{2}}$ of measurable function:     A function $f : X \to \mathbb R$ is measurable or $\mathcal A$-measurable if $\{x \mid f(x) &gt; a \} \in \mathcal A$ for all $a \in \mathbb R.$   To an untrained eyes, the problem with this problem is that there seems to be no problem at all, except, perhaps, that the exercise says "$\forall r \in \mathbb Q$", whereas the definition says "$\forall a \in \mathbb R.$" Am I on the right track here? Please help me going forward if this is the right issue to tackle.  Thank you for your time and help.    Footnotes:  (1) Richard F. Bass' , 2nd. edition, chapter 5: Measurable Functions, Exercise 5.1, page 44.  (2) Definition 5.1, page 37.
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.
How to choose capacitor for an IC I'm a beginner in electronics from the software industry. With some self-taught things, I'm trying to implement some basic Arduino circuits. My confusion is about capacitors mainly. My understanding about capacitors is that they act as power storage for few seconds or milliseconds.  I found that most IC's must have capacitors connected to their pins.  My confusion is how to find out which pins need a capacitor, and how to find the correct capacitor for a circuit or a capacitor for an IC.  Finally, why are capacitors necessary in a circuit in such situations? Strange error defining a macro with arguments for math mode I'm making a map where I want players to be able to choose there kits/classes. The problem I'm having with it though is giving players already enchanted Tools/Armour/Weapons.  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Thanks ! Should we MAC-then-encrypt or encrypt-then-MAC? Most of the time, when some data must be encrypted, it must also be protected with a , because encryption protects only against passive attackers. There are some nifty encryption modes which include a MAC (, ...) but let's assume that we are doing old-style crypto, so we have a standalone encryption method (e.g. AES with CBC chaining and PKCS#5 padding) and a standalone MAC (e.g.  with SHA-256). How should we assemble the encryption and the MAC?   MAC-then-Encrypt: Compute the MAC on the cleartext, append it to the data, and then encrypt the whole? (That's what  does) Encrypt-and-MAC: Compute the MAC on the cleartext, encrypt the cleartext, and then append the MAC at the end of the ciphertext? (That's what SSH does) Encrypt-then-MAC: Encrypt the cleartext, then compute the MAC on the ciphertext, and append it to the ciphertext? (In that case, we do not forget to include the initialization vector (IV) and the encryption method identifier into the MACed data.)   The first two options are often called "MAC-then-encrypt" while the third is "encrypt-then-MAC". What are the arguments for or against either? Open subscheme of an irreducible component remains open? Accepting an answer should automatically upvote it as well How To Get Notified When Hard Drives Fail In Dell PowerEdge Servers I have a Dell PowerEdge R415 and R515 and wondering how I can be alerted remotely when a hard drive fails in either server? Both are running RAID, so I need to be notified, so I can swap out the drive. Both servers only have the basic Baseboard Management Controller. Is this possible with a third party tool/software? I have a regular C# code. I have no exceptions. I want to programmatically log the current stack trace for debugging purpose. Example:  public void executeMethod()  {     logStackTrace();     method(); } What is considered a "real question"? Note:   This close reason is old and only applies to old questions that were closed prior to the closing overhaul. This has been replaced by the reasons "unclear what you're asking" for unclear questions and "too broad" for overly broad questions. Older questions that have been closed as "not a real question" will remain closed under this reason, so this question is left here for reference.   Questions used to be closed as "not a real question."     What makes a question a "real" question? What are new users expected to know about asking "real" questions? Why does SE treat questions differently from other online forums? In R, what does qqnorm actually do? qqnorm(x) plot(qnorm(seq(80)/81),sort(x))   Finding that the plots produced by the commands above are slightly different from each other, I tried this:  qqnorm(qnorm(seq(80)/81))   I got a slightly less than perfect line.  I'd have tried regressing the result of qnorm(seq(80)/81) on the variable that was plotted on the $x$-axis and plotting residuals against the predictor, expecting to see some graceful curve, but for the fact that I don't know what to use as the predictor.  Possibly such a residual plot would reveal more than just the graceful S-shaped thing I'd anticipate.  So my question is this: if the thing on the $x$-axis in the plot produced by qqnorm is not what I get from qnorm(seq(80)/81) (and what I did shows that indeed it is not), then what is it? Is it OK to ask on SO for help in explaining someone else's working piece of code, from Open Source projects or library examples etc.? Or does this belong somewhere else? 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The Academy, among other things, publishes a dictionary (""), in print and , which is usually given a lot of prestige (but is not without , of course).  Are there any such authorative—or at least influential—institution(s) or publication(s) for the English language? Minecraft Survival Mode Are there any official specifications regarding the torque for camera mounts? I've been wondering why I've never seen any official specifications of the maximum allowed torque that can be exerted upon lens mounts. It seems to me like a very useful specification, but I can't seem to find it from any manufacturer.  indicates that there might be no specifications for Nikon mounts. Are there really no manufacturer that publish these specification of their mounts?