Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions defined on a measure space $ (\Omega,\Sigma,\mu) $, where the index set $ A $ (assumed to be directed) is not necessarily $ \mathbb{N} $? That is, if   $ (f_{\alpha})_{\alpha \in A} $ converges pointwise almost everywhere to a measurable $ f $, there exists a measurable $ g $ such that $ |f_{\alpha}| \leq |g| $ almost everywhere for each $ \alpha \in A $, and $ \displaystyle \int_{\Omega} |g| ~ \mathrm{d}{\mu} \leq \infty $,   then does $ \displaystyle \lim_{\alpha \in A} \int_{\Omega} f_{\alpha} ~ \mathrm{d}{\mu} = \int_{\Omega} f ~ \mathrm{d}{\mu} $?
Let $G$ be a locally compact Hausdorff Abelian topological group.  Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure.  Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$.  Define the function from $G$ to $\mathcal{L}^1(G)$ that assigns to each $y$ in $G$ the function that takes $x$ in $G$ to $f(xy^{-1})$.  I.e. it assigns the $y$-translate of $f$.  Is this a continuous function of $y$ into $\mathcal{L}^1$?      On p.94 3.10 of  it is asserted that it is.  The usual proof of this sort of thing in cases where $G=\mathbb{R}$, say, is you use density of compactly supported continuous functions, and then dominated convergence.  The former survives, but dominated convergence does not work on nets.  In fact, a limit of a net of measurable functions need not be measurable.
If $X$ is distributed $N(\mu_X, \sigma^2_X)$, $Y$ is distributed $N(\mu_Y, \sigma^2_Y)$ and $Z = X + Y$, I know that $Z$ is distributed $N(\mu_X + \mu_Y, \sigma^2_X + \sigma^2_Y)$ if X and Y are independent.  But what would happen if X and Y were not independent, i.e. $(X, Y) \approx N\big( (\begin{smallmatrix} \mu_X\\\mu_Y \end{smallmatrix}) , (\begin{smallmatrix} \sigma^2_X &amp;&amp; \sigma_{X,Y}\\ \sigma_{X,Y} &amp;&amp; \sigma^2_Y \end{smallmatrix}) \big) $  Would this affect how the sum $Z$ is distributed? C++ Redefinition Header Files (winsock2.h) I want to make an arena where mobs will fight each other and a player can bet on one side or the other. For example: There are 2 skeletons in dyed armor; red and blue. Each color representing a team. When they spawn I want them to fight each and then when one dies I want it to announce which team is the winner and give the player the "money" due. I'm just not sure how to get the skeletons to attack each other without a player instigating it. Include custom taxonomy term in search I have two custom taxonomies applied to two custom post types. the terms list on the sidebar just fine and will list all posts associated with it. However, if you search one of the terms in specific, it doesn't bring up a post with that term.  Example:  Search for term "PQRI"  I get nothing. Any ideas? I've tried using various search plugins but they either break my custom search parameters or just don't work. I am a newbie to Drupal, and I really don't understand the difference between sub-theme and base theme. Why can't I directly edit the base theme?  I am working on a website which is live and has bartik as default theme. I don't know where to start. What is the difference between OpenOffice and the newly created LibreOffice? What is the difference between OpenOffice and LibreOffice? I understand that LibreOffice is another fork that is going to be developed. I have used OpenOffice since v1 and am wondering if there is a definite advantage to uninstalling and trying LibreOffice. Variable is already defined in method lambda Who are the Steam Mascots? So with the Steam Summer Sale in full swing, I've finally taken notice of the Steam Mascots.  Judging by the latest background, the black one, at least, has a name: Cyl.  Do the other two have names as well? Are there only these three?   What's their story? Association Bonus not removed upon account deletion Recently a user asked a question on Meta asking about his deleted account (I won't draw attention to him or his question unless it is really necessary but shouldn't be hard to find - at least until the question gets deleted1 as I don't know if it is relevant to this question).  However, while looking at his account, it appears as if he received the association bonus on all of his accounts, including the new Stack Overflow account he just created, but does not have any account with >200 rep.    The most likely scenario is that the deleted account had more than 200 rep and triggered  the association bonus, but that leads to 2 questions:   Why did his previously associated accounts retain the association bonus? Why did he get an association bonus on his brand new account?   Is this a  or ?  1 - The post in question was deleted by the OP several hours ago but before then the account was confirmed by a mod to be deleted due violations of a sock puppety nature (which I think everyone suspected at the time).  Given that the account in question was a sock puppet, I think it highlights an important reason for removing the association bonus when the account that triggered the bonus is deleted.  Theoretically, as long as he keeps getting the association bonus, he can go to any site and already have more than enough privileges to do damage. Every element of a group has order $2$. Why, intuitively, is it abelian? Blender 2.78 with Windows 10 lags when interacting with the UI I want to completely replace Ubuntu with Lubuntu The Barbarian's Rage ability gives advantage on Strength checks and saving throws.  Does this mean that the Barbarian gets advantage to attack rolls as well? Is an attack a check, or since saving throws are mentioned, that means it does not apply to attacks? I use the highlight mode in vim to copy a few characters.  I then want to paste more than once.  My current technique does not work well.  Sample text: Linux Solaris Irix HP-UX  Suppose I want to copy the word Linux, then paste over Solaris and Irix.   Place cursor at L in Linux Command v (for visual hilite), then e (for end-of-word), then y (for yank/copy) Now Linux is on my "vim clipboard" Move cursor to S in Solaris (first instance) Command v (for visual hilite), then e (for end-of-word), then p (for paste) Text is now: Linux Linux Irix HP-UX, but now Solaris is on my "vim clipboard" Move cursor to I in Irix (second instance) Command v (for visual hilite), then e (for end-of-word), then p (for paste) Text is now: Linux Linux Solaris HP-UX which is not what I expected.   I resort to using highlite/paste with the mouse (via X Terminal).  Surely, I can do this better.  How? Fast way of finding lines in one file that are not in another? Does Change Data Capture (CDC) work in a SQL Server 2012 AlwaysOn Failover Cluster Instances setup? Does  work in a  setup?  There would be two nodes in the AlwaysOn Failover Cluster Instances setup and two Domain Controllers.  We are not using AlwaysOn Availability Groups.  Will CDC work? and will it failover? For $n \geq 2$, prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$ For $n \geq 2$ prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  We need to use induction.   The Principle of Mathematical Induction, Theorem 4.2.1, states that $n_0 \in \mathbb{Z}$. For each integer $n \geq n_o$, let $P(n)$ be a statement about $n$. Suppose that the following two statements are true:   $P(n_o)$ $( \forall n \geq n_o)[P(n) \rightarrow P(n+1)]$   Then, for all integers $n \geq n_o$, that statement $P(n)$ is true.  Suppose,  $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$, the equation, is true for some natural number $k$.  $\frac{1}{2^2})*...*(1- \frac{1}{k^2}) = \frac{k+1}{2k}$  The induction hypothesis $P(k) \rightarrow P(k+1)$ must hold for any natural number $k, P(k),$ from the basis and inductive step.  $(\frac{1}{2^2})*...*(1- \frac{1}{k^2})*(1- \frac{1}{(k+1)^2}) = \frac{k+1}{2k}(1- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $\frac{k+1}{2k}(\frac{(k+1)^2}{(k+1)^2}- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $(\frac{(k+1)}{2k})( \frac{(k+1)^2-1}{(k+1)^2})$  $\leftrightarrow$ $( \frac{(k^2+2k+1-1}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{(k^2+2k}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{k+2}{(2)(k+1)})$  We need to demonstrate that   $(\frac{1}{2^2})*...*(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  holds true for $n \geq 2$  The question is do I just substitute a number that is greater than two after I did the induction? What is the Fastest a Character Can Move in One Turn? At level 20 with any magic items in the DMG, and any spells, what is the fastest a DnD 5e character could move using their movement?  (i.e., not including teleportation) theorem numbering: section counter but remove the section number I found this problem on a well-known problem solving website. It is apparently from Ramanujan.     With   $$\LARGE{a = \frac{1}{1 + \frac{e^{-2\pi}}{1 + \frac{e^{-4\pi}}{1 + \ddots}}}},$$   what is $a$?   The basic methods I know for questions of this type were not helpful to me.  I do not require a full solution, but feel free to post one if you would like. Hints or lecture notes on how to solve problems of this nature would be appreciated.   Keep in mind that I am at an undergraduate level in mathematics.