1

Attention

The usual problems are about absolute convergence: $$\int|g_n|\mathrm{d}\mu\quad(g_n=f_n,f-f_n,s_m-s_n,\ldots)$$

(There Fatou may help out!)

But as proceeding with Fatou one encounters that one can't distort to the limessuperior: $$\int\limsup_n|g_n|\mathrm{d}\mu\nleq\int\liminf_n|g_n|\mathrm{d}\mu\leq\liminf_n\int|g_n|\mathrm{d}\mu\leq\limsup\int|g_n|\mathrm{d}\mu$$

So the real question is about the analogue for limessuperior!!

(And not the analogue for negative functions...)

Problem

Given a measure space $\Omega$.

The lemma of Fatou states: $$f_n\geq0:\quad\int\liminf_nf_n\mathrm{d}\mu\leq\liminf_n\int f_n\mathrm{d}\mu$$ Does the reverse hold true: $$f_n\geq0:\quad\int\limsup_nf_n\mathrm{d}\mu\leq\limsup_n\int f_n\mathrm{d}\mu$$ Certainly, for convergent examples this holds true: $$f_n\geq0:\quad\int\limsup_nf_n\mathrm{d}\mu=\int\lim_nf_n\mathrm{d}\mu\leq\lim_n\int f_n\mathrm{d}\mu=\limsup_n\int f_n\mathrm{d}\mu$$ So one needs to dig deeper to find an honest counterexample!!!!

(I intend to answer my own question!)

(I have to admit that my earlier answer was lame!)

C-star-W-star
  • 16,275
  • 1
    what is the point of asking a question and posting the answer right afterwards ? – mookid Dec 12 '14 at 22:54
  • 5
    @mookid Can I answer my own question? Yes! Stack Exchange has always explicitly encouraged users to answer their own questions. If you have a question that you already know the answer to, and you would like to document that knowledge in public so that others (including yourself) can find it later, it's perfectly okay to ask and answer your own question on a Stack Exchange site. –  Dec 12 '14 at 22:57
  • @mookid: The point is to remind myself (and others). But please have a look on: Answer own Question – C-star-W-star Dec 12 '14 at 22:58
  • @mookid: You're welcome. ;) Unfortunately, many people don't ask about it before judging. :( – C-star-W-star Dec 12 '14 at 23:16
  • about your question: this is the whole point of the hypothesis $f\ge 0$ in the original Fatou lemma. – mookid Dec 12 '14 at 23:28
  • @mookid: That is not the only point; Fatou involves two choices: Either $f_n\geq0$ or $f_n\leq0$ as well as either $\liminf_nf_n$ or $\limsup_nf_n$. (Wikipedia discusses only one other choice namely what if it where say negative.) – C-star-W-star Dec 12 '14 at 23:31
  • Yes it is. Distinguishing $f_n\ge 0$ or $f_n\le 0$ is not a "choice". There is only one version, with $\liminf$ and $f_n\ge 0$ (or $f_n\ge -g$ with $g$ integrable, as it is trivially equivalent). – mookid Dec 12 '14 at 23:35
  • @mookid: You're right. ^^ (Taking limsup on negativ ones is just equivalent to the usual.) – C-star-W-star Dec 12 '14 at 23:37
  • @mookid: this sort of self-answered non-question is strongly discouraged on this particular SE site, although it is allowed by the software. – Carl Mummert Dec 13 '14 at 16:02
  • @Behaviour: answering your own question is permitted, but this particular kind has always been discouraged on math.SE. If a user discovers the answer after asking, then writing an answer is reasonable. If they know the answer before asking, there is no reason to ask. See http://meta.math.stackexchange.com/questions/4680/is-answering-own-question-okay for a longer discussion about a user who was asking questions only to answer them himself – Carl Mummert Dec 13 '14 at 16:04
  • @CarlMummert: We had this discussion already. The upshot was that you couldn't or didn't offer any reference for that!! – C-star-W-star Dec 13 '14 at 16:06
  • @Freeze_S: http://meta.math.stackexchange.com/questions/4680/is-answering-own-question-okay – Carl Mummert Dec 13 '14 at 16:06
  • @CarlMummert: Good, that's finally some reference. :) Anyway, in my opinion this thread is positive for stack exchange as wikipedia misses considering it and the only other post here containing this doesn't adress to directly to the positive nondominated case. Thus my proposal, leave it open! (Before, I always first try to find an answer via google, wiki and SE.) – C-star-W-star Dec 13 '14 at 16:17

2 Answers2

4

Attention

The problematic examples are the divergent ones: $\liminf\neq\limsup$

Counterexamples

On the one hand, it holds by the usual Fatou: $$f_n:=\chi_{(n,n+1]}:\quad0=\int\limsup_nf_n\mathrm{d}\mu\leq\limsup_n\int f_n\mathrm{d}\mu=1$$ On the other hand, it fails the reverse Fatou: $$f_{1\leq k\leq n}:=\chi_{(\frac{k}{n},\frac{k}{n}]}:\quad1=\int\limsup_nf_n\mathrm{d}\mu\nleq\limsup_n\int f_n\mathrm{d}\mu=0$$ (So while the former forces the direction of the inequality the latter shows it can't hold.)

C-star-W-star
  • 16,275
  • 1
    I don't understand your indexing of $f_n$ in $$f_{1\leq k\leq n}:=\chi_{(\frac{k}{n},\frac{k}{n}]}$$ – Duchamp Gérard H. E. Dec 29 '16 at 21:38
  • The ${1\leq k\leq n}:={(k,n):1\leq k\leq n}$ run as $(k,n)=(1,1),(1,2),(2,2),\ldots,(1,5),\ldots(5,5),(1,6),\ldots,(6,6),\ldots$ – C-star-W-star Dec 30 '16 at 10:44
  • So, in my language, it is $$g_n=\sum_{1\leq k\leq n}\chi_{[\dfrac{k}{n},\dfrac{k}{n}]}$$. Is what you want to define ? – Duchamp Gérard H. E. Dec 31 '16 at 06:13
  • No, the indexing here does not consist of only a single "number" $\lambda:=n$ but really of tuples of "numbers" $\lambda:=(nk)$. You can think of it is a diagonally counting the tuples similar to the proof that the rationals are countable: $1'=(11), 2'=(12), 3'=(22), 4'=(13),\ldots$. – C-star-W-star Dec 31 '16 at 15:04
  • 1
    In general, as you will hopefully learn later on, one is not limited to only take the natural numbers as indexing set (sequences) but any (directed) preordered set (net) will do a job too. This veeery often turns technical matters into natural ones. Also it comes in very handy in many constructions and applications as for example defining arbitrary sums/integrals. – C-star-W-star Dec 31 '16 at 15:10
  • Sorry to insist. If you have time, could you elaborate these mysterious notations (i) $f_{1\leq k\leq n}:=\chi_{(\frac{k}{n},\frac{k}{n}]}$, how can we compute $f(t)$ for $0\leq t\leq 1$ (ii) $1=\int\limsup_nf_n\mathrm{d}\mu\nleq\limsup_n\int f_n\mathrm{d}\mu=0$ then index $f_\lambda$ ($\lambda$, compositions ?). To end with, I agree that nets (together with subnets) are elegant for questions where non-linearly ordered (neither non-cofinal-countable) limits are involved (+1). – Duchamp Gérard H. E. Dec 31 '16 at 15:43
  • For example [this question][1] can be naturally (and elegantly) treated with nets and subnets. [1]:http://math.stackexchange.com/questions/515496/sum-of-closed-and-compact-set-in-a-tvs – Duchamp Gérard H. E. Dec 31 '16 at 16:39
  • Oh no worries I'm always in for conceptual math questions :D – C-star-W-star Dec 31 '16 at 23:34
  • Oh no no it is not always a good idea to use nets. In this particular MSE question nets may be a bad Ansatz. A strong indicator against nets in there is the ingredient compact set. Generally speaking, in any branch of mathematics matters become trivial in the right language. That is compactness becomes very natural in terms of neighborhood systems and filters! That is a subnet in terms of filters becomes simply set inclusion. However, I'll have to think about that particular MSE question.. – C-star-W-star Dec 31 '16 at 23:51
  • Let us continue this discussion in chat. – Duchamp Gérard H. E. Jan 01 '17 at 06:30
  • @Duchamp Gérard H. E. I don't understand this notation either what does indicator $f_{1\le k \le n} = \chi_{(n,n+1]}$means can you write the example such as $f_2$,$f_3$,$f_4$ in the integral? – yi li Dec 30 '19 at 06:55
2

Here's a simpler counterexample.

For $x \in [0,1]$, $f_{n}(x) = 1$ if $n$ is odd, and $0$ otherwise.

For $x \in (1,2]$, $f_{n}(x)=1$ if $n$ is even, and $0$ otherwise.

For any other $x$, $f_{n}(x) = 0$.

In this case $$ \int \limsup_{n} f_{n}(x) dx = 2,\\ \limsup_{n} \int f_{n}(x) dx = 1. $$

starboy
  • 351