Questions tagged [limsup-and-liminf]

For questions concerning limit superior and limit inferior of sequences (of real numbers and various generalizations) and also for $\limsup$ and $\liminf$ of sequences of sets.

Filter by
Sorted by
Tagged with
3 votes
0 answers

Convergence of the perimeter of level sets

I have already posted this question on Math StackExchange. Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}...
totallyimmersed9's user avatar
3 votes
1 answer

When are events in tail $\sigma$-algebra the limsup of some sequence of events?

Consider a sequence of $\sigma$-algebras $\mathcal{F}_1,\mathcal{F}_2,\dots$. Is it true that for any event $B$ in the tail $\sigma$-algebra $\mathcal{F_{\text{Tail}}}$, it can be expressed as the $\...
yellowello's user avatar
2 votes
1 answer

Understanding a proof about limit of a sequence of open sets

We are reading a proof about the following limit \begin{equation}\tag{1} \lim_{n \to \infty} \sigma_1(T_n)= \sigma_1(T), \end{equation} where $T:D(T) \subseteq H \to H$ and $T_n:D(T_n) \subseteq H \to ...
Maklen's user avatar
  • 229
4 votes
2 answers

Comparing two limsup's

Let $f\in L^2(0,\infty)$ be a positive, decreasing function. Is it then true that $$ \limsup_{x\to\infty} xf(x) = \limsup_{x\to\infty} \frac{1}{f(x)}\int_x^{\infty} f^2(t)\, dt $$ (and similarly for $\...
Christian Remling's user avatar
8 votes
2 answers

How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?

Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...
0xbadf00d's user avatar
1 vote
2 answers

Alternative characterization of epi-convergence

I am struggling with the proof of a property of epi-convergence. We need the following definitions: For a sequence of sets $(C^\nu)_\nu$ in $\mathbb R^n$, the outer limit is the set $\limsup_\nu C^\...
Philipp Wacker's user avatar
4 votes
0 answers

A subadditive bijection on the positive reals

I posed some time ago this question on MSE, which I am proposing also here since we got no definitive answer. Question. Does there exist a subadditive bijection $f$ of the positive reals $(0,\infty)...
Paolo Leonetti's user avatar