I am reading measure theory from Royden, and I am stuck in some of them. I have this question:      suppose  $E$ is  a measurable set and let $f: E \to \mathbb{R}$. Prove that : $f$ is measurable if and only if $f^{-1}(A)$ is measurable for any $A \subseteq \mathbb{R}$.   I know this is not true if measurable means "Lebesgue measurable", can anyone give a counterexample in details ?
Let $f$ be a continuous function on a set $E$.  Is it always true that $f^{-1}(A)$ is always measurable if $A$ is measurable?  I say no.  We know that $\bar{\psi(x)}:=\frac{\phi(x)+x}{2}$ where $\phi$ is the Cantor (or Cantor-Lebesgue) function and $\bar{\psi}:[0,1] \rightarrow [0,1]$.  We know this function maps a measurable subset of $C$ (the Cantor set) into a non-measurable set $W$.  A few observations about $\bar{\psi(x)}$.  It is strictly increasing and continuous, so it has a continuous inverse $\bar{\psi(x)}^{-1}$.  Thus $\bar{\psi(x)}^{-1}(W)=c \subset C$; that is, $\bar{\psi(x)}^{-1}(W)$ is non-measurable, but c is.  Is this correct?
How can I increment a date by one day in Java? I'm working with a date in this format: yyyy-mm-dd.  How can I increment this date by one day? How to use ampscript in order to pull user data from a second data extension? We have several data extensions that have data relationships with each other.  How can I send an email using one data extension while pulling data from a second data extension to personalize my email with ampscript?  I've tried Lookup in many different ways, but I've only managed to pull one specific row (the same row which is not personalized to each user) or all rows at once. I haven't been able to pull personalized data for each user.  In practice, I want to pull data from a DE called "Survey". Each user has a specific ID and some users have an "AnswerID", which varies. I want to to create IF/THEN statements based on that AnswerID.  Thanks, Sylvie Bending of light - photon's inertia instead of mass How do you safely encode a URL using JavaScript such that it can be put into a GET string?  var myUrl = "http://example.com/index.html?param=1&amp;anotherParam=2"; var myOtherUrl = "http://example.com/index.html?url=" + myUrl;   I assume that you need to encode the myUrl variable on that second line? How to prototype? How is prototyping done in a professional EE environment? Do you do your prototypes on breadboards, on copper clad boards, Manhattan style, or do you just draw schematics, order PCBs made and assembled and work with them, hacking here and there if necessary? How to disable autorefresh in snap Is it possible to disable autorefresh in snap? This  says that Snaps are updated automatically in the background to the latest version, every day. Some snaps (e.g. vlc) are built daily, and it makes no sense to install them every day. If $T^n$ is $q$-contractive, $T$ exactly has one fixed point Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one fixed point $\overline{x}\in X$.      The n-th power is defined inductively: $$ T^{n+1}(x):=T(T^{n}(x)), n\in\mathbb{N}. $$ And for a function $T\colon X\to X$, $q$-contractive means  $$ \exists 0\leq q&lt;1~\forall~x,y\in X: d(T(x),T(y))\leq q d(x,y). $$    Now to the proof.  I think I have to show, that $T$ is q-contractive, because then it follows with Banach, that $T$ does have exactly one fixed point. So I have to show, that there exists a $0\leq q&lt;1$ so that for all $x,y\in X$ it is $$ d(T(x),T(y))\leq q\cdot d(x,y). $$ And most likely I have to use the q-contractivity of $T^n$, i.e. that there exists a $0\leq q &lt;1$ so that for all $x,y\in X$ it is $$ d(T^n(x),T^n(y))\leq q\cdot d(x,y). $$  Can you help me? 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NB: I am making this question to provide a unified answer to questions about eigenvalues of small matrices so that all of the specific examples that come up can be marked as duplicates of this post. See . How do I move my website to a new server without interruption to email? If $\mathfrak p \subset R$ is a prime ideal, prove that for every nilpotent $r \in R$ it follows that $r \in \mathfrak p$.   The only hint that my tutor gave me was to use induction. Can someone explain what he means by this?  Thanks for the help!