Can we find a Borel measurable function $f:\mathbb{R} \rightarrow (0,\infty)$ such that $\int \mathcal{X}_I f d\mu = \infty$  for every nonempty open interval $I \subset \mathbb{R}$, where $\mu$ is the Lebesgue measure on $\mathbb{R}$ ? I think the answer is true; there is such a function, but with this condition &quot;integral = infinity&quot; this makes $f$ is unbounded almost everywhere, so not sure how to define it, if it exists.
Let $f : \mathbb R^n \rightarrow \mathbb R$ be a measurable function.  We know that $f$ is not necessarily locally integrable - but how hard can a measurable $f$ fail to be so? For example, is it true that $f$ is locally integrable outside of a negligible set $N \subset \mathbb R^n$?   Edit: In consideration of the comment by zhs, let me correct the phrasing of the question. I was originally wondering whether the 'singular' behaviour of a measurable function can be localized to a 'small' set.   A motivational example is $f(x) = 1/x$ over the real line. It is locally integrable in the complement of the origin. Hence, a conceivable phenomenon, which I had in mind originally when asking the question, could be that a measurable $f$ is integrable over open sets that are compactly contained in the complement of some negligible sets.
What happens to the second Marty at the end of Back to the Future? In  (the original movie), Marty comes back from 1955 10 mins early to prevent Doc from being shot (which he fails at). He then rolls up to the Lone Pine Mall and sees himself hit 88 mph and shoot back to 1955 (again). Marty then carries on with this life as usual. What happens to the Marty that goes back to 1955 at the end of the film?  This obviously is a recursive problem where it would create another timeline and Marty would keep coming back 10 mins early to view another Marty zip back to 1955? Rename Multiple files in unix I have multiple files whose names look like this  2313_ABCDEAUG2014_PQRST_0_.pdf 3244_ABCDEAUG2014_PQRST_0_.pdf 4234_ABCDEAUG2014_PQRST_0_.pdf   t want to rename the file in the following format.  AZ-2014-08-2313.pdf AZ-2014-08-3244.pdf   Please suggest. Say I have two Chrome windows open like this:   It seems that the most recently used window always appears on top. In other words, the &quot;Apply Digital Signatures&quot; was the window I had open last. If I were to switch to the window showing the new tab, then the order in which the windows are displayed in the task bar suddenly changes. Is there a way to keep them from shuffling around? I'd like to be able to order them however I want, like tabs in a web browser. (I'd also like to put them across different workspaces.) How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$? How do I root my Android device? This is a common question asked by those who want to root their phones. How exactly do I root my Android device? Are there risks involved? How to remove time from date? How to evaluate $\int_0^{2\pi} \frac{d\theta}{A+B\cos\theta}$? I'm having a trouble with this integral expression:  $$\int_0^{2\pi} \frac{d\theta}{A+B \cos\theta}$$  I've done this substitution: $t= \tan(\theta/2)$  and get: $\displaystyle \cos\theta= \frac{1-t^2}{1+t^2}$ and $\displaystyle d\theta=\frac{2}{1+t^2}dt$ where $\displaystyle \cos^2\theta/2=\frac{1}{1+t^2}$  then the integral becomes:  $$\int\frac{2 \, dt}{(A-B)t^2+(A+B)}= \sqrt\frac{A+B}{A-B} \arctan \left(\left(\sqrt\frac{A+B}{A-B} \right) t\right)$$   However, I'm not sure about the new limits since $\tan$ has period $\pi$ so what I have to do at this point to decide the new limits? And of course if you find some mistake in what I've done before, please let me know! Does it mean that two threads can't change the underlying data simultaneously? Or does it mean that the given code segment will run with predictable results when multiple threads are executing that code segment? I have come across both the phrases - 'on the internet' and 'over the internet.'   I'm pretty sure that both are correct but then which to use when?  Please note, I'm well aware of all over the Internet which I'm not concerned about. I would like to add in QGIS real time weather data.  I found this  and this  and I tried to follow the example, but without any success.  I've added a WMS weather data layer for precipitation to the QGIS map, as follows:  First, the parameters for the service:  service = 'crs=EPSG:900913&amp;dpiMode=7&amp;featureCount=10&amp;format=image/png&amp;layers=precipitation&amp;styles=&amp;url=http://wms.openweathermap.org/service'   Next, create the raster layer, specifying WMS as the type:  rlayer = QgsRasterLayer(service, "precip", "wms")   Finally, add the precipitation layer to the map:  QgsMapLayerRegistry.instance().addMapLayers([rlayer])   the example didn't show any error in the code and no data was added to the weather layer either, I'm not sure why? Why doesn't "its" have an apostrophe? I know that its is the possessive and it's is the contraction, and know when to use them.  But why doesn't the possessive have an apostrophe?   "The bear's eating a fish." [contraction] "The bear's coat is brown." [possessive] "It's eating a fish." [contraction] "Its coat is brown." [possessive] "One's eating a fish." [contraction] "One's coat is brown." [possessive]    lists the etymology as "From  +‎ ", and  says that this is actually the original form:     Originally written it's, and still deliberately spelled thus by some writers until early 1800s.   So what happened to the apostrophe?  When did people stop using it, and why did they?  It seems that it's as the possessive is more natural, as most people do this until they're taught that it is wrong (or even after).  Update: Online Etymology Dictionary has been updated to include two potential explanations:     The apostrophe came to be omitted, perhaps because it's already was established as a contraction of it is, or by general habit of omitting apostrophes in personal pronouns (hers, yours, theirs, etc.).   Can anyone back up either of these arguments?  The possessive one's still has the apostrophe, despite these. I want to prove that every real polynomial of odd degree has at least one real root, using  the intermediate value theorem.  Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \mathbb{R}$ and $n \in \mathbb{N}$.  By the fundamental theorem of algebra I know that $P(x)$ has exactly $2n+1$ complex roots, so  $P(x) = (x+r_1)(x+r_2) . . . (x+r_{2n+1})$ for each $r_i \in \mathbb{C}$  I do not know how to complete this but I do know that, at some point, I probably have to show that each root with imaginary part non zero has to come in conjugate pairs, and since $2n+1$ is odd there is at least $1$ root that is imaginary part $0$ and thus real. Extra Backslash Required in First Appended Newline in sed Is there a regular expression to detect a valid regular expression? pc ram memory 2x1GB or 1x2GB I don't know if it's the right place to ask but can somebody advise me for buying a ram memory for my computer.  I have a Dell Vostro 200 with 1GB of RAM and 667MHz. (2 x 512MB)  I want to buy raise the amount of RAM but I'm confused whether to buy 1 x 2GB or 2 x 1GB  I know it's the same for the available amount of memory but is ther a difference maybe in the clock speed so that 2 x 667MHz will be better than 1 x 667MHz. or maybe accessing one single memory slot will be faster and better than accessing two slots.  Please anyone helps. Number agreement between subject and object Is there a way to simulate multiple iphones using xcode/iphone sim? 2 orderby in wp_query with 2 custom fields I'm triyng to display hotels (taxonomy which i have created) by two 'orderby' with wp_query but i'm not success.  Results looks likes:  Hotel 1 :       2 persons          5 stars Hotel 2 :       2 persons          4 stars Hotel 3 :       2 persons          3 stars Hotel 4 :       3 persons          5 stars Hotel 5 :       5 persons          5 stars Hotel 6:        10 persons         2 stars Hotel 7:        10 persons         1 star   The first 'order by' is on the number of persons(the biggest in first) and the second 'order by' is on the rating in stars (so, if several hotels have the same number of persons, the hotel with the biggest stars rating would be in first).  Below my code. I have tried to add an array inside an array with meta_query but it doesn't work.  $args=             array(                                     'post_type' =&gt; 'hotel',                                     'Services' =&gt; $Services,                                     'Town' =&gt; $Town,                                      'meta_key' =&gt; 'Number_persons',                                     'orderby' =&gt; 'meta_value_num',                                     'order' =&gt; 'ASC',                                        'meta_query' =&gt; array(                                                            array(                                                                'meta_key' =&gt; 'Rating',                                                                'orderby' =&gt; 'meta_value',                                                                'order' =&gt; 'DESC',                                                               )                                                        )                  );   $the_query = new WP_Query( $args );   Thanks a lot :) Short Science Fiction story, told in a series of letters This was a short story told in a series of letters.  The protagonist starts by writing a letter to a utility company about an error in his bill.  By the last letter he is on death row trying to get his sentence commuted.  Each letter is slightly misunderstood.  I think it's from the late 60's and may have been in anthology like "Orbit". How to "add existing frameworks" in Xcode 4?