If $E$ has measure zero and $f$ is $C^1$, then $f(E)$ also has measure zero I was reading a proof that submanifolds $M^d \subset \mathbb R^n$ with $d&lt;n$ has measure zero. In the proof they use the fact: Let $f:U\longrightarrow\mathbb{R}^n$ $\mathcal{C}^1$ where $U$ is an open susbet of $\mathbb{R}^n$, then if $E$ is a borelian of null measure, then $f(E)$ is of null measure too. If I have this result, I know how to conclude. Can you help me showing this lemma? Thank you! Let $B$ be a closed ball of $U$ and $E$ a borelian of null measure, then since $f$ is $\mathcal{C}^1$, there exists a $K&gt;0$, such that $f$ is $K$-lipschitz continuous (by compacity of $B$ and by mean value inequality). We conclude that the image by $f$ of a cube $C$ of measure $\delta&gt;0$ is a set of measure at most $K^n\delta$. For all $\varepsilon&gt;0$, we can cover $E\cap B$ by an union of cube $C$ such that $\lambda_n(C)\leq\varepsilon$. I don't understand this one since we don't think we have compacity of $E\cap B$... I just understood that we can obtain a $f(C)$ arbitrarily small if we take a cube $C$ arbitrarily small.
Why does a Lipschitz function $f:\mathbb{R}^d\to\mathbb{R}^d$ map measure zero sets to measure zero sets? Why does a Lipschitz function $f:\mathbb{R}^d\to\mathbb{R}^d$ map measure zero sets to measure zero sets?   It is easy to prove this statement if the domain is bounded. Is there any way to extend the argument to unbounded domains? Can anyone give me some hint, or show me a simple proof sketch?
I am using \if condition in a custom command in my LaTeX document. I would like to know where I can find some good explanation about \if command. I followed a  but it's just an example; there are not any explanation. Do AWS Reserved Instances need to be assigned to the instance? I have an RDS instance running of type db.t2.small.  I bought a reserved instance for 1 year of the same type, I paid the partial upfront rate.  Do I need to assign this subscription in some way to my existing instance to say 'This is the instance that is using the subscription' or does AWS just charge for the instance usage depending which instances you have running and whether an available subscription was active for it at the time which determines the running rate?  The reason that I ask is that I can't see any mention of the reduced rate on my AWS Billing area and it shows that I've used 17 hours of my reservation, but there isn't a row on my bill so far this month that indicates the new partial upfront rate that I should be charged for my instance. Is LAN with StarCraft 2 definitely not possible? Word or phrase for making something seem better by comparison What is a word or phrase that expresses the idea of purposely making something seem better by comparison? For example, lemon Starburst was created to make the cherry flavor seem that much tastier. Often confusion arises when the  spell is used, an  is worn, or the 's anti-magic ray is employed, whether a given character's ability will still function.   The spell does give you some guidelines on how to adjudicate the effects of the loss of magic but says very little about how to determine what is affected.  In short, how do I tell if an ability is magical? How to exit early on pipe close? I use golang for writing programs that are too complex to express as bash functions. The programs read stdin line by line and write modified output to stdout.  The problem with this iterating over all the lines in stdin approach, is that it doesn't exit early. For example, the following command will process all the lines instead of just 10:  cat largefile | custom-program | head -n 10   How do the native unix application manage to exit early and how can I do the same in my custom scripts? How do I analyze a .hprof file? Why doesn't C# support implied generic types on class constructors? C# doesn't require you to specify a generic type parameter if the compiler can infer it, for instance:  List&lt;int&gt; myInts = new List&lt;int&gt; {0,1,1,     2,3,5,8,13,21,34,55,89,144,233,377,     610,987,1597,2584,4181,6765};  //this statement is clunky List&lt;string&gt; myStrings = myInts.     Select&lt;int,string&gt;( i =&gt; i.ToString() ).     ToList&lt;string&gt;();  //the type is inferred from the lambda expression //the compiler knows that it's taking an int and  //returning a string List&lt;string&gt; myStrings = myInts.     Select( i =&gt; i.ToString() ).     ToList();   This is needed for anonymous types where you don't know what the type parameter would be (in intellisense it shows up as 'a) because it's added by the compiler.  Class-level type parameters don't let you do this:  //sample generic class public class GenericDemo&lt;T&gt;  {     public GenericDemo ( T value )      {         GenericTypedProperty = value;     }      public T GenericTypedProperty {get; set;} }  //why can't I do: int anIntValue = 4181; var item = new GenericDemo( anIntValue ); //type inference fails  //however I can create a wrapper like this: public static GenericDemo&lt;T&gt; Create&lt;T&gt; ( T value ) {     return new GenericDemo&lt;T&gt; ( value ); }  //then this works - type inference on the method compiles var item = Create( anIntValue );   Why doesn't C# support this class level generic type inference? How do I convert an argument list to a sequence of arguments? Non-empty intersection. Let $A_1,A_2,...,A_{2019}$ be sets s.t. $|A_i|$=45 for each $i=\overline{1,2019}$ and $|A_i\cap A_j|=1$ for every $i\neq j$. Is it true that $$\Big|\bigcap _{i=1}^{2019} A_i\Big|=1?$$ What I thought of doing is proving by induction by i, the base is obvious. However I'm stuck on the next step. Explicit bijection between the successor of $\alpha$ and $\alpha$ (infinite ordinal). I would like to construct an explicit bijection from $\alpha^*$ to $\alpha$, where $\alpha$ is an infinite ordinal. I have been told a possible bijection would be:  \begin{equation} f(\beta) = \begin{cases} 0 &amp; \text{if } \beta = \alpha \\ \beta^* &amp; \text{if } \beta \text{ is finite} \\ \beta &amp; \text{otherwise} \end{cases} \end{equation}  I don't understand how this is constructed, or why it is a bijection. Evaluate $\sum_{n=1}^\infty nx^{n-1}$ How can you evaluate $\sum_{n=1}^\infty nx^{n-1} = \frac{1}{(1-x)^2}$ without relying on the fact that it's the derivative of $\sum_{n=1}^\infty x^n = \frac{1}{1-x} $? Why cannot fermions have non-zero vacuum expectation value? Is it possible to install Ubuntu without formatting any partitions? Object appears smoothed after rendering but not in viewport If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$? Why are we scared of singularities? I often hear people say that general relativity predicted its own demise because of the singularities it predicts. If I'm not mistaken, this is also a problem in QFT. I wonder why singularities garner so much hate. To me, it seems like negative or complex numbers: we used to hate these things but now they are more generally accepted. Why can't it be that nature just has singularities? We might not get it, we might not like it, but it may be so. How do Zombies differentiate between a zombie and a human being? Zombies are always after human beings. How do Zombies differentiate between a zombie and a human being? Why don't zombies eat each other? Where to get school zone shapefiles? Census.gov usually makes available the shapefiles containing school zone boundaries. Due to the US government shutdown, census.gov is down. Does anyone know where to download a recently archived copy? How should an educational institute woo good faculty?