Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{A}$ be an algebra of subsets of $\Omega$ such that  $\sigma(\mathcal{A})=\mathcal{F}$. Prove that for all $B\in \mathcal{F}$  and for all $\varepsilon &gt;0$ there is $A\in \mathcal{A}$ such that $P(B\bigtriangleup A)&lt;\varepsilon,$ where $B\bigtriangleup A:=(A\setminus B)\cup (B\setminus A)$ is the symmetric difference of $A,~B.$  I have tried proof by contraposition, by assuming the existence of a set $B \in \mathcal{F}$ and an $\varepsilon&gt;0$ such that $P(B\bigtriangleup A)\geq \varepsilon,$ but this doesn't seem to get me somewhere. On the other hand, for a straight proof, the construction of the desired set $A$ seems foggy, since I don't know how to start.   Thanks a lot for the help!
Theorem. Let $(X,\mathcal B,\mu)$ a finite measure space, where $\mu$ is a positive measure. Let $\mathcal A\subset \mathcal B$ an algebra generating $\cal B$. Then for all $B\in\cal B$ and $\varepsilon&gt;0$, we can find $A\in\cal A$ such that $$\mu(A\Delta B)=\mu(A\cup B)-\mu(A\cap B)&lt;\varepsilon.$$  I don't think there is a proof in this site. It's a useful result for several reasons:  We know what the algebra generated by a collection of sets is, but not what the generated $\sigma$-algebra is. The map $\rho\colon \cal B\times\cal B\to \Bbb R_+$, $\rho(A,A')=\mu(A\Delta A')$ gives a pseudo-metric on $\cal B$. This makes a link between generating for an algebra and dense for the pseudo-metric. We say that a $\sigma$-algebra is separable if it's generated by a countable class of sets. In this case, the algebra generated by this class is countable. An with the mentioned result, we can show that $L^p(\mu)$ is separable for $1\leq p&lt;\infty$, which makes a link between the two notions. In ergodic theory, we have to test mixing conditions only an a generating algebra, not on all the $\sigma$-algebra.
How do I disable X at boot time so that the system boots in text mode? Addons: Only some .zips will install Why is it that if I go to install an add-on from User Preferences > Add-ons > Install from File... and choose a *.ZIP, some add-ons will install, while others will return me to the User Preferences with nothing having happened (as far as I can tell)?  Is there a specific file structure required in the .ZIP? I get many add-ons from GitHub, using the Download Zip button, and even there installation of the actual Zip file only works some of the time. Where to set proxy ignore list? I upgraded to ubuntu 11.10 from 11.04, shocked to see few things...    My system is behind a proxy, so I did set a proxy setting. Being a web developer, I want to remove a lot of addresses that I type in browser address bar to not hit proxy. There's no such ignore list in ubuntu 11.10, what/where do I do it? What was the requirement to change the way network gui worked? When is tension constant in a rope? Consider a massless rope with pulling forces applied at each end.  How do we decide if tension is constant or not in a rope? Consider a few example scenarios:  For example, if there is a knot in the rope the tension is not constant throughout (why?. Similarly, if the rope is hung over a cylindrical pulley of a non-negligible radius, the tension is not constant (why?). If there isn't anything touching the rope, for example, 2 people tugging on the rope at each end, the tension is constant (why?). Very strange error suddenly appeared running Biber I am on a Mac with MacTex 2020 and ran into a strange problem with Biber. Working on a document all was fine, I did not change the bibliography (says my git) and only fixing typos (not even adding cites) and suddenly latexmk stopped working after a Biber run, while that looked like  INFO - This is Biber 2.14 INFO - Logfile is 'doc.blg' INFO - Reading 'doc.bcf' INFO - Found 31 citekeys in bib section 0 INFO - Processing section 0 INFO - Globbing data source 'literature.bib' INFO - Globbed data source 'literature.bib' to literature.bib INFO - Looking for bibtex format file 'literature.bib' for section 0 INFO - LaTeX decoding ... INFO - Found BibTeX data source 'literature.bib'   so harmless in my view. Then with --debug I additionally get in the end.  INFO - LaTeX decoding ... warning: overriding existing definition of macro "jun" warning: overriding existing definition of macro "jul" warning: overriding existing definition of macro "sep" warning: overriding existing definition of macro "jan" [...] warning: overriding existing definition of macro "nov" INFO - Found BibTeX data source 'literature.bib' Unicode::UCD: failed to find unicore/version in /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc CODE(0x7fd026aeb5a8) CODE(0x7fd026aebb90) at /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib/Biber/Input/file/bibtex.pm line 1011.   It seems unrelated to the library itself since deleting a few entries (going through deleting one half, then the other), the problem persists. It does not happen for an empty literature, but as soon as a first entry is added (just author/journal/title/year/volume/pages - no utf8-only chars) this error message appears.  what does this last error message mean?  How can I solve this? How do candles and wicks work? Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n This is my first question on here. I'm trying to numerically solve the Schrödinger equation for the  and find the energy eigenvalues and eigenfunctions but I am confused about how exactly this should be done.  I've solved some initial value problems in the past using iterative methods such as Runge–Kutta. I've read that  is the way to solve Schrödinger's equation but Wikipedia also describes it as an iterative method for initial value problems.  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PART 1: I was reading the article  Wikipedia. I couldn't understand this line:       "if the two events are causally connected ("event A causes event B"), the causal order is preserved (i.e., "event A precedes event B") in all frames of reference."     Is this an assumption or a consequence of STR? Please explain.       Note: My question consists of 2 parts this is the 2nd part. Below is a genral version of my previous question question:.   PART 2: Let there be three events $A$,$B$ and $C$ s.t: $C$ is the result of Simultaneous occurrence of $A$ and $B$. In other words $C$ occurs iff $A$ and $B$ are simultaneous. Now as we know in STR any two events separated in space are not simultaneous in different frames. So In some frames $C$ will occur and in some $C$ will not occur which will cause paradox. I tried many thought experiments to make such a paradox but i failed. In all the experiments that i thought i could not break the causality even by breaking the simultaneity because everytime the fact: "all signals move slower than light" preserved the causality.    So why causlity remains preserved always? Is it due to the fact that nothing can move faster than light? For one of the clients, I have spin up a modern communication site collection and on the landing page of the same, I am seeing 'Get the mobile app' showing up, I was able to hide the 'feedback' button using css but no luck with Get the mobile app button.  Can someone please share css to hide 'get mobile app' button, appreciate the help. Why isn't smoke flowing with the shape of the domain? I created this Smoke domain.    Then When I added smoke, it didn't stay within the domain, it went through the hole. reflecting direct light on glossy surface in Cycles Working in Cycles.  Say I have a "disco ball" with a glossy shader. I'm pointing a spot light to it, and I'm trying to have the reflections of that light illuminate the diffuse material on the walls with the patterns from the ball.    I get this:    but I'd like to get something like:    For the illustration I placed a point light inside the ball an removed faces. But what I'd like to do is move the light around the ball so that the light reflected from it changes like it would do in real life.   Is that possible at this point?  EDIT:  Just for the record here's the image rendered in LuxRender (yet another program to learn...) How do I get started creating an Ubuntu Desktop App? "The volume boot has only 0 bytes disk space remaining" After a recent update, I'm getting an alert saying:  The volume boot has only 0 bytes disk space remaining   But my computer has plenty of HD space free. Does anyone know how I resolve this. (If it's relevant, I'm using the whole disk encryption feature of the alternate install image for Ubuntu 12.04). I want to remove all the files from different directories, and want to keep only latest files 20 files. is this the correct command to do this?  ls -t1 /mnt/dwh/ftp/dwh_ftp_cbs/ARLOGS/ | tail -n +22 | xargs rm -f How is my laptop ram running in dual channel? So essentially does margin collapsing occur when you don't set any margin or padding or border to a given div element? Is there a way to disable WAC String for a specific link (ex. Tracking Link) with AMPscript or otherwise? URLs: Should I use hyphens, underscores or plus symbols? Which is better for search engines?   example.com/my_cool_page.html example.com/my-cool-page.html  example.com/my+cool+page.html How can I  compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?