How to prove that cardinality Borel $\sigma$-algebra equals the cardinality of $\mathbb R$? My understanding at this point is that to assign a probability measure to a random variable defined on the real line, we need a Borel $\mathscr{B}$ sigma algebra, because otherwise we wouldn't be able to find a bijection from every subset of the power set ($2^\Omega$) to the Lebesgue measure, due to the fact that the cardinality of $2^\Omega$ is not just greater, but much greater than the subsets of the Borel algebra.   On the other hand, the Borel sigma algebra, even though it contains every number in the real line as a singleton set, has that same cardinality as $\mathbb{R}$.  How can this be proven? Is there a way to see this through the concept of standard topology?
cardinality of the Borel $\sigma$-algebra of a second countable space Second countability by itself doesn't restrict the cardinality of a topological space, since every set with the trivial topology is a second countable space, but it seems natural to ask whether second countability restricts the cardinality of the Borel $\sigma$-algebra of the space.  Can the cardinality of the Borel $\sigma$-algebra of a second countable space be arbitrarily big? If this is the case is there a simple construction for a second countable space with an arbitrarily big Borel $\sigma$-algebra?
I have the following problem: Let $(M, \omega)$ be a symplectic manifold. How can I show $$\omega^n=\underbrace{\omega\wedge \ldots\wedge \omega}_{n-times},$$ satisfies $\omega^n(p)\neq 0$ for all $p\in M$. I believe that is not too dificult but I'm not used with exterior product.. Any help will be valuable...  This is important because the nondegeneracy condition on $ω$ is equivalent to the condition that M has an even dimension $2n$ and the top wedge product $\omega^n$ is nowhere vanishing on M, i.e., $ω^n$ is a volume form. In particular, $M$ must be orientable and is canonically oriented by $\omega^n$. How to create this curved 90-degree-angle quickly I know I need to have (although I don't know why) a GROUP BY clause on the end of a SQL query that uses any aggregate functions like count, sum, avg, etc:  SELECT count(userID), userName FROM users GROUP BY userName   When else would GROUP BY be useful, and what are the performance ramifications? Has confinement been experimentally observed? So, confinement has obviously been shown by lattice gauge theory to be a predicted aspect of QCD. However, to what extent has it been observed in experimental physics? Where in the DNS process are NS records used? I register a new domain, mynewdomain.com, at a hosting company.   In the advanced DNS settings there is an A record which maps mynewdomain.com to a ip address.  In my webhost control panel, it shows me the nameservers for the domain: ns1.mywebhost.net &amp; ns2.mywebhost.net  When I type in mynewdomain.com into a browser and hit enter I believe the following happens.   ISP resolver searches "." nameserver ISP resolver then told to search ".com" nameserver instead. ISP resolver finds mynewdomain.com, and searches the zone file to get the A record.     So my question is, where does my webhost's nameservers come into play? I believe the flaw in my logic comes from the fact that my zone file (in which the A record is kept) is stored on ns1.mywebhost.net). But if that is the case, how does my ISP resolver know to look here? (because I didn't type this into my address bar) Displaying two XML listings side by side. I have two XML outputs from a program that I would like to include in a LaTeX document. I would like them to appear side-by-side each in its own box. So that a reader can compare the two outputs while reading the document. Can't seem to find a way to do this, without making images for each output and adding both as images/graphics. Could there be a better option? How to quickly and safely defrost chicken? What is the safest, quick way to defrost chicken? Microwave? Running water over it? How to create Motion Trail in Cycles I want to create a motion trail effect for a sword like known in games look at that example if you have an solution that looks not as fancy as in the picture don't mind to post it. I'll accept anything useful. Is there any good reason that an empty set of round brackets (parentheses) isn't valid for calling the default constructor in C++?  MyObject  object;  // ok - default ctor MyObject  object(blah); // ok  MyObject  object();  // error   I seem to type "()" automatically everytime. Is there a good reason this isn't allowed? Why Can't I duck reverb on the mix bus in Unity? i'm surprised there is so little documentation on this issue. But i'm trying to duck the audio on all the mix buses in Unity so that you can clearly hear the VO of the game.  Now it works great except that I have some reverb zones in the game and any sound that plays in these reverb zones, the actual reverb itself is NOT DUCKED.  So the VO is accompanied with a huge wall of wishy washy muddy noise.  Anyone deal with this before? How can we deal with this? List partition labels from the command line Is there a command that will list all partitions along with their labels? sudo fdisk -l and sudo parted -l don't show labels by default.  EDIT: (as per comment below) I'm talking about ext2 labels - those that you can set in gparted upon partitioning.  EDIT2: The intent is to list unmounted partitions (so I know which one to mount). How do I remove the limit on PC keyboard button presses? Whenever I play a PC game with the directional keys, I can only press two buttons at a time.  For instance if I am playing a shooter like R-Type, I cannot hold down the fire button while moving diagonally at the same time.  The computer simply ignores the extra button press.  How do I remove this limit so I can shoot while moving diagonally?  This limitation seems to apply for every game.  I am running Windows 7 if it makes any difference.  I noticed the same problem back when I used Windows XP. Calculate $\int_{0}^{1} (x-f(x))^{2016} dx$, given $f(f(x))=x$. Show community wikis differently in the Android app We sometimes buy trout from a local breeder who'll get them fresh out of the water for us, and then cook them a few hours later. They taste great, but the flesh tends to fall apart in a way that doesn't happen with fish from the super market. It's often barely even possible to get them out of the pot without them completely coming apart. We tried leaving them in the fridge for a day so they wouldn't be quite as just-killed-an-hour-ago-fresh, but that didn't help.   We usually prepare them by cooking them in simmering (not boiling) water.  Is there some way to prevent this? Available US State or National business (or employer) database lists? This question dovetails with this post on .  I am familiar with some of larger, paid-for business registry's like  and .  While the detailed information their business listings database include (like # of employees, annual sales, industry code, etc) is fantastic, if you are on a limited budget, sometimes the cost of purchasing the data outweighs the value you can get from it.  If you have geocoding capabilities in house, one could potentially create a reasonable surrogate by spatially-enabling (geocoding) a publicly available business listing on your own.    If you were looking to take on this type of project, where would you start your research? Do you know of US agencies at a State or National level that maintain this type of information, AND are publicly available for download or minimal purchase? Deprecated: Function split() is deprecated. How to rewrite this statement? "Congratulate for" vs. "congratulate on" Which is correct?     I congratulated him for coming first in the race.      I congratulated him on coming first in the race. Cancel/kill window.setTimeout() before it happens I have a few places where I use the line below to clear out a status, for example.  I have a few of these that hang out for 10 seconds or more and if the user gets clicking around the action can occur at incorrect time intervals.  window.setTimeout(function() { removeStatusIndicator(); }, statusTimeout);   Is it possible to cancel or kill this with some jQuery or JavaScript code, so I don't have this process hanging around? How do I connect the initial highway to my six-lane road?