Tagged Questions

49 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
23 views

How to prove $f(x)=e^{\frac{1}{x}}$ is continous in $(0,a), a>0$ and $\int_{0}^{a}e^{\frac{y}{x}}dx, y>0$ does not exist

I would aprecciate any advice. I'm trying to prove that in the context of a measure space, $(X,B,\lambda)$ , with $X=(0, + \infty)$, $B$ the Borel sigma-algebra and $\lambda$ the Lebesgue measure, ...
55 views

Continuous function bounded in $L^\infty$
Is a continuous (real-valued) function in $L^\infty$ a (everywhere-)bounded function?
Using Lusin's Theorem to show that continuous functions are dense in $L^p$
Lusin's theorem says that in a finite measure space, given a measurable function $\varphi$, for every $\varepsilon \gt 0$ there exists a continuous function $g$ such that  \mu\left(\{x : ...