How to I prove that f is measurable if f is a real function of a measurable space $X$ such that $\{x ; f(x) \geq r\}$ is measurable for every rational $r$.  From my basic understanding I need to prove that $f^{-1}(U)$ is measurable where U is the union of countably open balls with rational radii?
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 do I identify which features have NULL attributes in the attribute table of a shapefile?  When i try [field]= 'NULL' it selects the features with value = 0 (see picture) The  mention that it is possible to add pie menus. But I'm not sure how to do it.  How could I convert the mesh select mode menu's appearance:    to this? How can I sort arrays and data in PHP? I want to change my field from molecular biophysics to theoretical ecology. While they are basically required some same basic background (physics, math, programming), the rest is quite irrelevant. In my current draft of this part, about a half is about this the problem and the result of the work, another half is for my contribution and what I've learnt. Having said that, should I focus less on describing the work and leave space for the thing that will be useful for the next research?  Below is copy from .     What have you already done? What problems have you solved, or at least worked on? What independent projects have you been part of? What were your key contributions? What did you learn? What did you teach the world? How do your results compare to what was already known? What original ideas are you most proud of? Be specific, technical, credible, and confident (but not arrogant). Refer the reader to your web page for more details. Have a web page with more details: preprints, project reports, source code, videos, etc. Proving the measure of an increasing sequence of measurable sets is the limit of the measures Show that if $A_1\subseteq A_2\subseteq A_3\subseteq\cdots$ is an increasing sequence of measurable sets (so $A_j\subseteq A_{j+1}$ for every positive integer $j$), then we have $$m\left(\bigcup_{j=1}^\infty A_j\right)=\lim_{j\to\infty}m(A_j)$$  Here is my proof: According to the $\sigma$-algebra property, $\bigcup_{j=1}^{\infty}A_j$ is a measurable set, so it makes sense to talk about $m(\bigcup_{j=1}^{\infty}A_j)$. Firstly, I prove that $\lim_{j\to\infty}m(A_j)\leq m(\bigcup_{j=1}^{\infty}A_j)$. This is because for any given positive integer $N$, $A_N\subseteq \bigcup_{j=1}^{\infty}A_i$, according to monotonicity, we have $m(A_N)\leq m(\bigcup_{j=1}^{\infty}A_i)$. Take the limit,we will have $\lim_{j\to\infty}m(A_j)\leq m(\bigcup_{j=1}^{\infty}A_j)$. Secondly, I prove that $m(\bigcup_{j=1}^{\infty}A_j)\leq \lim_{j\to\infty}m(A_j)$. For any given positive integer $N$, $\bigcup_{j=1}^N A_j = A_N$. According to monotonicity,we have $m\left(\bigcup_{j=1}^N A_j\right)=m(A_N)\leq \lim_{j\to\infty}m(A_j)$. Take the limit, we will have $m\left(\bigcup_{j=1}^\infty A_j\right) \leq \lim_{j\to\infty} m(A_j)$. Combine the above two arguments, we will see that $$m\left(\bigcup_{j=1}^\infty A_j\right)=\lim_{j\to\infty} m(A_j)$$$\Box$  The above is my proof, unlike many books, my proof does not use the property of countable additivity. So I doubt my proof is correct. Who can point out where are my mistakes? Cardinality of a set of closed intervals What is the cardinality of the set S of all closed intervals on the real number line with rational (positive) lengths?  I believe the number of intervals with a specific fixed length but varying start points is |R|, and the number of intervals with a fixed start point but varying lengths is |Q| = |N|. I think the answer to the question would then be |S| = |R x N| = |R|? I'm not quite sure how to prove the last equality though. Would I have to define a bijective function that maps elements of (R, N) to R? How would I approach doing so? The pdf of $\frac{X_1-\bar{X}}{S}$ Suppose $X_1, X_2,...,X_n$ be i.i.d from $N(\mu,\sigma^2)$ with unknown $\mu \in \mathcal R$ and $\sigma^2&gt;0$  Let $Z=\frac{X_1-\bar{X}}{S},$ S is the standard deviation here.  It is can be shown that  $Z$ has the Lebesgue p.d.f.  $$f(z)=\frac{\sqrt{n} \Gamma\left(\frac{n-1}{2}\right)}{\sqrt{\pi}(n-1)\Gamma\left(\frac{n-2}{2}\right)}\left[1-\frac{nz^2}{(n-1)^2}\right]^{n/2-2}I_{(0,(n-1)/\sqrt{n})}(|Z|)$$  My question is then how to get this pdf?  The question is from  in example 3.3.4  to find the UMVUE of $P(X_1 \le c)$. I can understand the logic and procedures to find the UMVUE but don't know how to get the pdf.  I think this question also relate to this    Thank you very much for help or point to any related references will be also appropriated. I am using setInterval(fname, 10000); to call a function every 10 seconds in JavaScript. Is it possible to stop calling it on some event?   I want the user to be able to stop the repeated refresh of data. What is the difference between Gödel's completeness and incompleteness theorems? Cleaning my Macbook Pro Retina leaves greasy 'streaks' across the screen I try to clean my mid-2015 Macbook Pro with Retina display by wiping the screen with a damp cloth and then immediately drying.  However, there seems to be left some residue/streak marks, which look very oily. Sometimes these disappear after a few minutes, but they are still very unsightly, and a bit worrying (is there something wrong with my screen?)  I'm wondering if there are other cleaning techniques which avoid this issue, or whether this is completely normal behaviour.  Thank you! How did the English language come to this?       The play was awful.     Is the complete opposite of     The play was awesome.   But if you break it down to awe followed by ful or some, it doesn't make sense at all. Can someone shed a light on this?  Is there a rule to this, or is it one of those things you need to just memorize? How does password salt help against a rainbow table attack? I'm having some trouble understanding the purpose of a salt to a password.  It's my understanding that the primary use is to hamper a rainbow table attack.  However, the methods I've seen to implement this don't seem to really make the problem harder.  I've seen many tutorials suggesting that the salt be used as the following:  $hash =  md5($salt.$password)   The reasoning being that the hash now maps not to the original password, but a combination of the password and the salt.  But say $salt=foo and $password=bar and $hash=3858f62230ac3c915f300c664312c63f.  Now somebody with a rainbow table could reverse the hash and come up with the input "foobar".  They could then try all combinations of passwords (f, fo, foo, ... oobar, obar, bar, ar, ar).  It might take a few more milliseconds to get the password, but not much else.  The other use I've seen is on my linux system.  In the /etc/shadow the hashed passwords are actually stored with the salt.  For example, a salt of "foo" and password of "bar" would hash to this: $1$foo$te5SBM.7C25fFDu6bIRbX1.  If a hacker somehow were able to get his hands on this file, I don't see what purpose the salt serves, since the reverse hash of te5SBM.7C25fFDu6bIRbX is known to contain "foo".  Thanks for any light anybody can shed on this.  EDIT:  Thanks for the help.  To summarize what I understand, the salt makes the hashed password more complex, thus making it much less likely to exist in a precomputed rainbow table.  What I misunderstood before was that I was assuming a rainbow table existed for ALL hashes. Creating non-reverse-engineerable Java programs Is there a way to do an "upside down" \widehat? I've come across this in a book I'm looking through, but I can't find the code for it. I want it to look exactly like the \widehat and go above text like \widehat does...I just want it to be (vertically) flipped! I want to do something like the following in TeX:  \begin{nobreak}    Text here will not split over pages, it will remain as one continuous chunk. If there isn't enough room for it on the current page a pagebreak will happen before it and the whole chunk will start on the next page.    \end{nobreak}   Is this possible? What books, journals, and electronic resources are most valuable for expanding knowledge of GIS? What books, journals, and/or electronic resources have you found most valuable for expanding your knowledge in the GIS field, and why? How to match all occurrences of a regex Landscape topography from grayscale image While using Wild Shape, do I use Unarmored Defense or the creature's Armor Class? Is $n^{th}$ root of $2$ an irrational number? Possible Duplicate:          Will every $n^{th}$ root of $2$ be an irrational number? If yes, how can I prove that?