Intereting Posts

Calculating sin and cos based on combination of exponentiation and power series?
How many solutions possible for the equation $x_1+x_2+x_3+x_4+x_5=55$ if
Prove that if $R$ is an integral domain and has ACCP, then $R$ has ACCP
Zero locus of a maximal regular sequence in closure of smooth, quasiprojective variety
Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$
How do you solve the cauchy integral equation?
Topologist's sine curve is not path-connected
How to prove that if each element of group is inverse to itself then group commutative?
If a set is compact then it is closed
injective and surjective
Game Theory and Uniform Distribution question?
Easy explanation of analytic continuation
Taylor series expansion of $\log$ about $z=1$ (different branches)
Why do knowers of Bayes's Theorem still commit the Base Rate Fallacy?
Proof of Non-Ordering of Complex Field

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$.

In order to prove this, for each $n \in \mathbb{Z}$ let $E_n=E \cap (n,n)$. How to show that the regularity theorem for $E_n$ can be solved entirely within the interval (n,n+1). And then re-include $\mathbb{Z}$ get to the end of the proof.

- Proving that the terms of the sequence $(nx-\lfloor nx \rfloor)$ is dense in $$.
- What is integration by parts, really?
- for infinite compact set $X$ the closed unit ball of $C(X)$ will not be compact
- Prob. 9, Chap. 6, in Baby Rudin: Integration by parts for improper integrals
- Open Cover for a Compact Subset
- If $f(0) = 0$ and $|f'(x)|\leq |f(x)|$ for all $x\in\mathbb{R}$ then $f\equiv 0$

- Picking a $\delta$ for a convenient $\varepsilon$?
- Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$
- How to find the limit of the sequence $x_n =\frac{1}{2}$, if $x_0=0$ and $x_1=1$?
- An example of a function not in $L^2$ but such that $\int_{E} f dm\leq \sqrt{m(E)}$ for every set $E$
- References on integration: collections of fully worked problems (and explanations) of (1) advanced and (2) unusual techniques
- $\lim_{x\to 0+}\ln(x)\cdot x = 0$ by boundedness of $\ln(x)\cdot x$
- What is the domain of $x^x$ when $ x<0$
- Using Leibniz Integral Rule on infinite region
- Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c)$ where $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k<c<y_k$
- Metric triangle inequality $d_2(x,y):= \frac{d(x,y)}{d(x,y)+1}$

We first have to deal with the technicality of what is meant by ‘Lebesgue measure’ as there are several, but equivalent, ways of defining it. I’m assuming the Lebesgue measure to mean the completion of the Borel measure on the reals, where the Borel measure is defined to be the restriction of the outer measure induced by the obvious pre-measure on the algebra generated by sets of the form $(a,b]$. With these assumptions the measure of a Lebesgue-measureable set $E$ is: $$\lambda(E)=\inf\{\sum_{j=1}^\infty(b_j – a_j):E\subseteq\bigcup_{j=1}^\infty(a_j, b_j]\}$$

Now we need a slight lemma which states that

$$\lambda(E)=\inf\{\sum_{j=1}^\infty(b_j – a_j):E\subseteq\bigcup_{j=1}^\infty(a_j, b_j)\}$$

Note the only change is that the cover in the second is composed of open intervals while the definition requires covers of half-open intervals. This lemma should be clear: countable sets have Borel measure zero so we can addend the right-hand ends to the cover of the second to get a cover consistent with the first definition. And we can take a cover consistent with the definition and surround the right-hand ends by ‘tiny’ open intervals to get a cover of open intervals. Taking the infimum of covers like this yields the same measure.

With this lemma we automatically get that for every Lebesgue-measure set $E$ and any $\epsilon$ there is an open set $U$ such that $E\subseteq U$ and $\lambda(U)\leq \lambda(E)+\epsilon$ (just choose $U$ to be the union of the open intervals that cover $E$). To get that there is closed set $K$ such that $K\subseteq E$ and $\lambda(E)\leq \lambda(K)+\epsilon$ breaks into two case. When $E$ is bounded we consider the set $\overline{E}-E$ (which is measurable) and use the fact that there is an open set $U$ such that $U\supseteq \overline{E}-E$ and $\lambda(\overline{E}-E)\leq \lambda(U)+\epsilon$. Setting $K=\overline{E}-U$ procures us with our desired interior closed (in fact compact) set. When $E$ is unbounded we set $E_j=E\cap (j,j+1]$ and apply the previous case to each of them getting a sequence of $K_j$s. Although it is not in general true that a countable union of closed sets is closed, it is true in this very special case of pairwise disjoint closed subsets of the reals. Now if $\cup K_j$ isn’t within $\epsilon$ of your $E$ you go back and do some fancy choosing of the $K_j$s to ensure that $K_j$ approximates $E_j$ to within $\epsilon 2^{-j}$ (note that I’m implicitly using an enumeration of the integers by the natural numbers which you can make more rigorous). Now we have that for any measurable subset $E$ and $\epsilon>0$ there is an open set $U$ and a closed $K$ such that $K\subseteq E\subseteq U$ with $\lambda(U)-\epsilon\leq \lambda(E)\leq\lambda(K)+\epsilon$.

With this fact in hand the problem becomes easy. Choose a sequence of $U_j$ and $K_j$ such that $\lambda(U_j)-1/j\leq \lambda(E)\leq \lambda(K_j)+1/j$. Setting $H=\cup K_j$ and $G=\cap U_j$ procures us our $F_\sigma$ and $G_\delta$ sets. Showing that $\lambda(G-H)=0$ can be seen by the fact that $\lambda(U_j-K_j)\leq 2/j$. Thus considering the intersection of the $U_j-K_j$ (which is $G-H$) has measure $0$.

- If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $$ is divisible by 56 but not 49.
- Does this group presentation define a nontrivial group?
- How is (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ ¬r) ≡ (p ∧ ¬r) ∨ (¬q ∧ q)? Is it really distributive property?
- If a prime can be expressed as sum of square of two integers, then prove that the representation is unique.
- Why is the expected number coin tosses to get $HTH$ is $10$?
- Compute $ \sum_{k=1}^{\infty} \text{sech}(2 k)$
- Range of A and null space of the transpose of A
- Do isomorphic quotient fields imply isomorphic rings?
- Divergence of the serie $\sum \frac{n^n}{n!}(\frac{1}{e})^n$
- If $$ and $$ are relatively prime, then $G=HK$
- Area of intersection between 4 circles centered at the vertices of a square
- Calculating the median in the St. Petersburg paradox
- The existence of partial fraction decompositions
- Infinite solutions of Pell's equation $x^{2} – dy^{2} = 1$
- Lower hemicontinuity of the intersection of lower hemicontinuous correspondences