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Show that $\lim\limits_{y\downarrow 0} y\mathbb{E}[\frac{1}{X};X>y]=0$. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $X$ be a nonnegative random defined on this space. For any $A\in \mathcal{F}$, let $\mathbb{E}[X;A]:=\mathbb{E}[X\mathbb{1}_{A}]$, where $\mathbb{1}_{A}$ denotes the indicator random variable on $A$. Without assuming $\mathbb{E}\left[\frac{1}{X}\right]&lt;\infty$, show that  \begin{eqnarray} \lim\limits_{y\downarrow 0} ~~y~\mathbb{E}\left[\frac{1}{X};X&gt;y\right]=0. \end{eqnarray} Any initial ideas will be greatly appreciated. Check if url has domain name and not an ip Prime ideal in a polynomial ring over an integrally closed domain Can a child enter the UK on an accompanied visa and then leave the country alone? My daughter who is just under 18 has a child accompanied visa to the UK. I am traveling with her to the UK for a few days. We return from UK on the same day except that she returns to India and I am heading to San Francisco an hour later on the same day.   Is that okay or will we have an issue at immigration into the United Kingdom? Sqlite3 by default sorts only by ascii letters. I tried to look in google, but the only thing I found were informations about collations. Sqlite3 has only NOCASE, RTRIM and BIARY collations. How to add support for a specific locale? (I'm using it in Rails application) Use of "save" or "save for" in sentences The use of "real" in the following cases Possible Duplicate:             If you listen real close...      Can you swing by real quick...   Sentences like the above two are what I often hear in daily life. If I didn't hear them in the real world, I would probably be more ready to say "If you listen really close..." and "Can you swing by really quick...".  Is real being used as an adjective in this scenario? If so, is the usage of adj.+ adj. a common practice in English language? How to make traditional output for derivatives monitor files (à la tail -f) in an entire directory (even new ones) I normally  watch many logs in a directory doing tail -f directory/*. The problem is that a new log is created after that, it will not show in the screen (because * was expanded already).  Is there a way to monitor every file in a directory, even those that are created after the process has started? A Russian girl is asking for 500 euros to get a passport in Russia. Is this a realistic amount of money for a passport fee? How to create a torus with divided cuts that correspond to the direction of the torus When moving large directories using mv, is there a way to view the progress (%)? The cp command on gentoo had a -g switch that showed the progress. Let, $X$ be a countable set. Which of the following are true?   There exists metric $d$ on $X$ so that $(X,d)$ is complete. There exists metric $d$ on $X$ so that $(X,d)$ is not complete. There exists metric $d$ on $X$ so that $(X,d)$ is compact. There exists metric $d$ on $X$ so that $(X,d)$ is not compact.   4 is true as we can take discrete metric. Then $\{\{x\}:x \in X\}$ is an open cover containing no finite subcover.  If there exists a compact metric then it would automatically become complete also. Hence if 3 is true, so is 1.  What about 2? Any other comment? Java socket/serialization, object won't update A folder was created in my home folder (not by anyone), and I cannot see the folder contents. She has root privileges, but when the nautilus root access the folder is empty. When I try to remove it, it will appear to remove more than 1 million files from my computer, and I do not know what these files are.  How i can see these files? the name of folder created is "4F8ZBREShQ". I'm scared about it... How to get straight quotation marks? How do I get straight quotation marks, i.e., " and ' in TeX?  (No, \verb changes the font and this is unacceptable.)  [Edit]  I forgot to mention that I’m using XeLaTeX, and fontspec is a necessity.  The first two proposed answers did work, without fontspec. What is LookDev/Material Preview mode for in 2.8? Custom ordering of duplicate targets is not necessarily honored on mobile apps Properties of $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$ where $\{a_n\}$ is unbounded, strictly increasing sequence of positive reals Bash Script if condition is always true