Intereting Posts

General Information about Eigenvalues for an 3×3 symmetric matrix
Tensors constructed out of metric other than the Riemann curvature tensor
Special arrows for notation of morphisms
Finding the infinite series: $3 \cdot \frac{9}{11} + 4 \cdot \left(\frac{9}{11}\right)^2 + 5 \cdot \left(\frac{9}{11}\right)^3 + \cdots$?
How to define the operation of division apart from the inverse of multiplication?
If a plane is divided by $n$ lines, then it is possible to color the regions formed with only two colors.
Alternative construction of the tensor product (or: pass this secret)
Pigeonhole principle application
A 10-digit number whose $n$th digit gives the number of $(n-1)$s in it
Finding for every parameter $\lambda$ if matrix is diagonalizable
Find the number of seven-letter words that use letters from the set $\{\alpha,\beta,\gamma,\delta, \epsilon\}$…
Dushnik-Miller Proof
$A \subset \mathbb{R^n}$, $n \geq 2$, such that $A$ is homeomorphic to $\mathbb{R^n} \setminus A$, and $A$ is connected
Are monomorphisms of rings injective?
Product of logarithms, prove this identity.

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.

- Is $x^{-0}$ defined?
- Continuously Differentiable Curves in $\mathbb{R}^{d}$ and their Lebesgue Measure
- Prove that if a sequence $\{a_{n}\}$ converges then $\{\sqrt a_{n}\}$ converges to the square root of the limit.
- Show that $f(x) = 0$ for all $x \in $ given $|f'(x)| \leq C|f(x)| $
- Converging series question, Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then does $\sum_{n=1}^{\infty} \frac {a_n}{n}$
- Spreading points in the unit interval to maximize the product of pairwise distances

- What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
- Do continuous linear functions between Banach spaces extend?
- Gaussian-like integral : $\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$
- All derivations are directional derivatives
- Limit of solution of linear system of ODEs as $t\to \infty$
- $\lim_{x\to 0+}\ln(x)\cdot x = 0$ by boundedness of $\ln(x)\cdot x$
- A question about continuous curves in $\mathbb{R}^2$
- Two problems on real number series
- Continuity of one partial derivative implies differentiability
- What will be a circle look like considering this distance function?

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 $0$.

- What are Quantum Groups?
- Compare growth rate of functions
- homomorphism of Laurent polynomial ring
- Any more cyclic quintics?
- Compute $\sum\limits_{n=0}^{\infty}\frac{1}{(2n+1) . 4^n} $
- Gradient in differential geometry
- Determining the Orbits/Orbit Space of $O(3)$ on Real 3 by 3 Traceless Symmetric Matrices
- How can I get a irreducible polynomial of degree 8 over $Z_2$?
- How is $\displaystyle\lim_{n\to\infty}{(1+1/n)^n} = e$?
- About the Fermat quotients with base $2$
- Surjectivity of a map between a module and its double dual
- Integral $\int_0^{\pi/4} \frac{\ln \tan x}{\cos 2x} dx=-\frac{\pi^2}{8}.$
- Linear Algebra: If $A^3 = I$, does $A$ have to be $I$?
- Existence of a meromorphic functions $f(z)$ such that $|f(z)|\geq |z|$.
- Least squares fit for an underdetermined linear system