2,731 reputation
425
bio website
location
age
visits member for 2 years, 2 months
seen 41 mins ago

33m
comment A sebset of $\Bbb C^2$
Suppose $f\colon\mathbb C^2\to\mathbb R,(z,w)\mapsto\lvert zw\rvert$. Note that if $I$ is an interval such that $m(I)$ is small, then $f^{-1}(I)$ is measurable and $m(f^{-1}(I))$ is also small.
38m
comment How to prove this question? a question about Riemann integration!
Note that $\max(f,g)=(f+g+\lvert f-g\rvert)/2$.
41m
comment Outer measure and Caratheodory's criterion
@GiuseppeNegro Generally, if $\mathcal A$ is a collection of subsets closed under countable intersection, then there's $A_0\in\mathcal A$ such that $m^*(A_0)=\alpha=\inf_{A\in\mathcal A} m(A)$. We again choose $A_n$ such that $m^*(A_n)\ge\alpha+1/n$, then let $A_0=\bigcap_n A_n\in\mathcal A$ and by def $m^*(A_0)\ge\alpha$, but $m^*(A_0)\le m^*(A_n)\le\alpha+1/n$.
49m
comment Outer measure and Caratheodory's criterion
@GiuseppeNegro $\inf$ in my expression is really $\min$, i.e. if my expression of $m^*(A)$ is right, then it should be a regular measure, since suppose $m(E_n)\le m^*(A)+1/n$ and $E_n\supseteq A$, then consider $E=\bigcap_n E_n$. Since $E\supseteq A$, we have $m(E)\ge m^*(A)$, but $m(E)\le m(E_n)\le m^*(A)+1/n$, thus $m(E)=m^*(A)$.
11h
reviewed Approve suggested edit on How do I count ordered tuples in this inequality?
11h
revised Book suggestions on projective geometry
added 44 characters in body
11h
comment Proof about Number Fields
Note (elementarily) that if $0\neq f\in\mathbb Z[X]$ and $m/n\in\mathbb Q$ is irreducible, then $m$ divides $f(0)$ and $n$ divides the leading coefficient of $f$.
16h
revised Book suggestions on projective geometry
added 2 characters in body
16h
revised Book suggestions on projective geometry
added 4 characters in body
16h
asked Book suggestions on projective geometry
17h
comment Outer measure and Caratheodory's criterion
@GiuseppeNegro Good. I took the wrong book so I didn't reach the same content. Now I borrowed the right book and found that the measure satisfying the condition I listed in the original post is just so-called regular outer measure in Munroe's book. Please post an answer filling the details and I'll accept your answer.
Apr
13
comment Proving that a function has a removable singularity at infinity
Comment on the last paragraph: as hinted in the preceding exercise of Ahlfors, when $f$ is entire, $f(z)=o(\lvert z\rvert)$ follows from Shwarz integral formula.
Apr
13
comment Outer measure and Caratheodory's criterion
@GiuseppeNegro Did you mean Munroe's GTM89 Introduction to Measure Theory and Integration? I've looked up and found nothing about this, but a discussion on the agreement of premeasure and the corresponding outer-measure.
Apr
12
comment Is there a simple proof for Fundamental theorem of finitely generated abelian group?
Try Serge Lang's Algebra or Michael Artin's Algebra.
Apr
11
reviewed Approve suggested edit on The general term formula $a_{n+1}=\dfrac{1+a_n^2}2$
Apr
11
answered Is convex hull of a finite set of points in $\mathbb R^2$ closed?
Apr
11
comment Formula for the following sum?
Such an algorithm exists for hypergeometric version of anti-difference. Gosper-Zeilberger's algorithm. See, for example, here.
Apr
11
comment I'm not able to solve the following indefinite integral
They are so-called elliptic integrals.
Apr
11
comment Bounded entire function constant
And additionally, $g(z)\to0$ as $z\to\infty$ by condition, thus $g(z)=0$.
Apr
11
revised Existence of a non-singular $n-2$ principle minor
deleted 6 characters in body