Intuition behind Caratheodory's definition for measurable sets? I start saying that I know this question has been already proposed a thousand times on this forum BUT I couldn't find a direct answer so I try asking on my own.  What's the intuitive meaning of this definition of measurable set?  $\cdot$ Let $X$ be a set and $\lambda^*$ an outer measure. Then $X$ is said to be measurable if for each $B\in X \lambda^*(B)=\lambda^*(B\cap X) + \lambda^*(B\cap X^C)$  Or equivalently if  $\lambda_*(X)=\lambda^*(X)$.  I feel OK with the second definition because it looks kind of similar to the intuition in Riemann Jordan "measure", but why are the two equivalent?  Thanks.
Intuition behind the definition of a measurable set This week I saw the definition of a measurable set for an outer measure.     Let $\mu^*$ be an outer measure on a set $X$. We call $A \subseteq X$ measurable if $$\mu^*(E) = \mu^*(A\cap E) + \mu^*(A^c\cap E)$$ for every $E \subseteq X$.   This is not the first time I've seen this definition. Unlike most other things in mathematics, over time I have gained absolutely no intuition as to why this is the definition.   The only explanation I've ever seen is that a set is measurable if it 'breaks up' other sets in the way you'd want. I don't really see why this is the motivation though. One reason I am not comfortable with it is that you require a measurable set to break up sets which, according to this definition, are non-measurable; why would you require that? Of course, you can't say what a non-measurable set is without first defining what it means to be measurable so I suppose no matter what your condition is, it will have to apply to all subsets of $X$.     Is there an intuitive way to think about the definition of measurable sets? Is there a good reason why we should use this definition, aside from "it works"?
Induced metric on a null hypersurface Consider a metric $g_{\mu\nu}(x)$ and a hyper surface ${\cal H}$ defined by $~f(x) = c$. One (or at least I) usually finds the "induced metric" on ${\cal H}$ by solving$~f(x) = c$ for one of the coordinates and plugging it back into the metric to give us $\gamma_{ij}$. For null hypersurfaces, I seem to be finding that the resultant induced metric is singular.         Is it true in general that the induced metric as defined above is singular on a null hypersurface?   How does one define the induced metric on a null hypersurface? Calculate the sum of inverse values of ${n\choose 0}, {n\choose 1}, ... {n\choose n}$ Calculate $$A={1\over {n\choose 0}}+ {1\over {n\choose 1}}+ ...+{1\over  {n\choose n}}$$  and  $$B={1\over {n\choose 0}}- {1\over {n\choose 1}}+ ...+{(-1)^n\over  {n\choose n}}$$     My idea for $A$ is some probabilistic reasoning. Color the sets $$\{\}, \{1\}, \{1,2\}, \{1,2,3\},...\{1,2,...,n\}$$ and ask our self what is the probability that I choose colored set among all sets. Clearly this is exactly ${n+1\over 2^n}$ and on the other side it is $A$:  probability that I take empty set is ${1\over {n\choose 0}}$,   probability that I take colored set with 1 element is ${1\over {n\choose 1}}$  probability that I take colored set with 2 elements is ${1\over {n\choose 2}}$  and so on ...   So $A ={n+1\over 2^n}$. But I have no idea how to attack $B$. Assume $f$ and $g$ are differentiable at $x$. Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$   I am assuming here $fg = f(x) g(x)$. Then we can prove this via induction. If $n = 0$ we have $1 = 1$ which is true. Now assume it is true for some $m$ we need to show it is possible for $m+1$. We have $(fg)^m = \sum_{k=0}^m f^{(k)}(x)g^{m-k}(x)$. How do we show that? How can I re-enable backspace navigation in Chrome? Armature keeps spinning around root bone in pose mode when I try to move anything I was following a tutorial on rigging (was pretty much done) and now my entire model keeps spinning around the main root bone if I try to move any bone at all. (first picture is where it originally was and still is while in Rest Position, second image is of how it changes when in Pose Position while in Pose Mode or Object mode, Edit Mode seems fine)    I tried clearing All Transforms or hitting the ALT+g, s, r while all were selected and that just caused it to spin in the same direction like everything else does as shown in this picture:   I was nearly finished with this video tutorial here,     and near the last part my issue began. 2:28:07 was the last time I was in edit mode and the pose was fine. Right after the little turrets he then went and parented 3 more bones and then he switches back to pose mode with no problem, I suspect somewhere in there is where the problem occurred on my end. I could have sworn I followed it correctly but maybe I went wrong somewhere I have no idea.   I couldn't find another issue like mine exactly so sorry if this question has been asked before, I'm still very new.  Edit: I checked in the video comments to see if it had happened to anyone else but I did not see any mention of it unfortunately.  Edit2: Picture for more detail on the yellow lines. Find polynomials $A(x)$ and $B(x)$ such that $A(x)P(x) + B(x)Q(x) = x + 1$ for all $x$ where $P(x) = x^4 - 1$ and $Q(x) = x^3 + x^2$. I'm stumped on this question. I know that I'm supposed to apply the extended version of Euclid's algorithm for polynomials but I'm unsure of how to do that. I thought about trying to create some kind of linear system but guessing arbitrary coefficients but that wouldn't work as $A$ and $B$ don't have fixed degrees. What system monitoring tools are available? I am looking for system monitoring tools which are GUI and CLI or web-based which include basic functions such as:   CPU Usage Ram Usage Swap Usage Disk Usage ( Space / I/O ) Heat Monitoring   I know there are many tools I can use, but I am looking for a single tool that has these basic functions. Imagine you are surviving in the wilderness traveling in a wandering manner. You've found this landscape that is wonderfully abundant in food and basking in a very favorable climate. Therefore, you decide that this is your new residence.  Night is approaching, so you haphazardly piece together a makeshift shelter to make for a more enjoyable living experience while you get your act together.  This shelter is only temporary. After you've become familiar with the land and have gathered finer resources, you will begin creating a proper home complete with securities, amenities, and furnishings.  The word I am looking for is an adjective meaning something similar to "created recklessly for temporary use."      The shelter you were building on your first night was a(n) ________   abode.   The words and phrases "ad-hoc", "off the cuff", "on the run" came to my mind, but none of those are understood to mean what I am trying to communicate (in fact, two of them are not even close). If you think of a word with a meaning that you believe to be similar to what I am searching for, I'd love to hear from you!  Thank you. What is god for religious people? In order to discuss with religious friends about god, I would like to know if there's some accepted objective definitions of God for main religions, even if belief is subjective.   For example, is he "a person who..."? In that case I would address features of a person, as capabilities and limitations. Is he "an entity..."? In that case I would use systems theory.   I feel lost on this, being agnostic from a religious family, where I was taught that God is a mess of things, including a beautiful bearded face, omnipotence and a mix of human defects and metaphysical powers. Direction of friction for object that rolls with slipping Gnome terminal will not start I just tried to install python 3.6 on my Ubuntu 16.04 system, and now I can't run the terminal from the launcher or from Ctrl + Alt + T. I tried to run gnome-terminal from XTerm and got the following message:  Traceback (most recent call last):   File "/usr/bin/gnome-terminal", line 9, in &lt;module&gt;     from gi.repository import GLib, Gio   File "/usr/lib/python3/dist-packages/gi/__init__.py", line 42, in &lt;module&gt;     from . import _gi ImportError: cannot import name '_gi' Error in sys.excepthook: Traceback (most recent call last):   File "/usr/lib/python3/dist-packages/apport_python_hook.py", line 63, in apport_excepthook     from apport.fileutils import likely_packaged, get_recent_crashes   File "/usr/lib/python3/dist-packages/apport/__init__.py", line 5, in &lt;module&gt;     from apport.report import Report   File "/usr/lib/python3/dist-packages/apport/report.py", line 30, in &lt;module&gt;     import apport.fileutils   File "/usr/lib/python3/dist-packages/apport/fileutils.py", line 23, in &lt;module&gt;     from apport.packaging_impl import impl as packaging   File "/usr/lib/python3/dist-packages/apport/packaging_impl.py", line 23, in &lt;module&gt;     import apt   File "/usr/lib/python3/dist-packages/apt/__init__.py", line 23, in &lt;module&gt;     import apt_pkg ModuleNotFoundError: No module named 'apt_pkg'  Original exception was: Traceback (most recent call last):   File "/usr/bin/gnome-terminal", line 9, in &lt;module&gt;     from gi.repository import GLib, Gio   File "/usr/lib/python3/dist-packages/gi/__init__.py", line 42, in &lt;module&gt;     from . import _gi ImportError: cannot import name '_gi'   How can I fix this error?  It should also be noted that:   My gnome-terminal file is now a python script. The gnome-terminal.real file will open the terminal as expected. python3.5 gnome-terminal will open the terminal as expected. Let $F,G,H$ be groups such that $F\trianglelefteq G \trianglelefteq H$.  I am asked whether we necessarily have $F\trianglelefteq H$. I think the answer is no but I cannot find any counterexample with usual groups. Is there a simple case where this property is not true? Let $X$ be adapted integrand, for which we know that $E[\int_0^T X_s^2 \, ds]&lt; \infty $. Let $0≤s&lt;t≤T$ and we have some $W∈F_s$ ($F_s$ is $\sigma$-algebra) and we know $E[\int_s^t W^2X_u^2 \, du]&lt;\infty$. I have to show  that $\int_s^t W X_u \, dB_u=W \int_s^t X_u dBu$.  I know that for simple integrands this follows from definition. We also know that for $X$ there exists sequence of elementary integrands $X^n$ such that $E[ \int_0^t (X^n_u-X_u)^2 \, du]$ converges to $0$, when $n \to \infty$. From this one can see that it also holds $E[ (\int_0^t (X^n_u-X_u) \, dB_u)^2]$. I wonder if I can show that it  holds $E[ |(\int_s^t W (X^n_u-X_u) \, dB_u)|] \to 0$ as $n \to \infty$ . (I think I should use Cauchy- Schwarz inequality: $E(XY)^2 \leq E(X^2)E(Y^2)$, but not sure where. If I could show this convergence in $L^1$, it would be enough for my proof. If I rewrite it as $E[E[|(\int_s^t W(X^n_u-X_u) \, dB_u)|  \mid F_s]]$ I don't know if I can put $W$ out of integral. Would it be correct? Any help would be really very appreciated. Thanks. If the dot product between two vectors is $0$, are the two linearly independent? If we have vectors $V$ and $W$ in $\mathbb{R^n}$ and their dot product is $0$, are the two vectors linearly independent?  I can expand $V_1 \cdot V_2 = 0 \Rightarrow v_1w_1+...+v_nw_n = 0$, but I don't understand how this relates to linear independence. Custom Edit control inside a ExtJS Editor grid How to enable touchpad? Yesterday, suddenly my touchpad on my MSI Laptop started to not respond. In other words, in Login screen, touchpad is working as expected. However, in Desktop screen, after I have logged in, touchpad is not working.  How can I enable touchpad in Desktop? How does the ternary operator work? How add 14 days on end date I have End Date column and I want to add 14 Days on it auto when created, can I do it by workflow? Can I do it via jQuery. If yes how? if no how I can do it? What are the alternatives to OS X's Spotlight? How many arguments does a function require, and how to use that in `Manipulate`