Every measurable function is the limit a.e. of a sequence of continuous functions. I already proved the finite-valued case i.e. any measurable $f:A\to\Bbb R$ can be approximated by a sequence of continuous functions a.e. Now suppose $f:A\to\Bbb [-\infty,\infty]$: measurable and consider a trunc...

$F$ is a field such that for every $a\in F, a^4=a$, then what is the characteristic of $F$? Take any $a,b \in F-\{0\}$. then $(a+b)^4=a+b\implies a+b +4a^3b+6a^2b^2+4ab^3=a+b\implies 4a+4b+6a^2b^2=0$. Multiplying throughout by $ab$ and using $a^3=b^3=1$, we get $4a^2+4b^2+6=0$. I am not sure how ...

Suppose there are distinct primes $p,q$ such that $\bar{f}=gh\in F_p[x]$，$\bar{f}=uv\in F_q[x]$, where $\bar{f}$ is the image of $f$ under natural projection from $\mathbb{Z}[x]$ to $F_p[x], F_q[x]$ respectively. If $g,h,u,v$ are irreducible in their respective rings, and $\{deg(g), deg(h)\}$ as ...

I read through some of John Gabriel's the new calculus last night, he does seem quite arrogant but some of the things he says seems to make sense. For example is his way of taking derivatives based of parallel secant lines right? To me it just seems like rolls theorem. Is this guy correct with h...

In Problems and Solution in Mathematics by Ta-Tsien, exercise 5123, the mean value theorem is used as: \begin{equation} \text{log} |F(0)| = \frac{1}{2 \pi} \int_0^{2\pi} \text{log}|F(re^{i\theta})| d\theta \end{equation} Considering the expression provided by http://en.wikipedia.org/wiki/Harmon...

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EDIT: As 5pm points out below, this is of course actually good enough since every domain in $\mathbb{C}$ is locally simply connected and harmonicity is a local property! This was too long for a comment, but I thought it might be nice to know. There is a much nicer, less computational way to pro...

Given assumption: Given a fiber bundle $\pi:E\rightarrow B$ with fiber $F$. Let $\phi:U\times F\rightarrow \pi^{-1}(U)$ and $\psi:V\times F\rightarrow \pi^{-1}(V)$ be local trivializations of $\pi$. Suppose locally, so on $V\times F\rightarrow V$, for every local trivialization $\Psi$ and compact...

$f: D\to \mathbb C$ is holomorphic such that $\sup_{u,v\in D}|f(v)-f(u)|=d$, then $|f'(0)|=d/2\implies f$ is linear. Here, let's suppose that $D$ is a closed unit disk. By Cauchy's integral formula, since $f(z)$ and $f(-z)$ are both holomorphic, we have for every circle $C_r$ centered at $0$ and ...