2
$\begingroup$

I need help with the following question:

Suppose $(X, \mathcal{S}, \mu)$ is a measure space with $\mu(X) < \infty$. Suppose $f_1,f_2,\dots$ is sequence of $\mathcal{S}$-measurable functions from $X$ to $\mathbb{R}$ such that $\lim\limits_{k \rightarrow \infty} f_k(x) = \infty$ for each $x \in X$. Prove that for every $\epsilon > 0$, there exists a set $E \in \mathcal{S}$ such that $\mu(X \setminus E) < \epsilon$ and $f_1,f_2,\dots$ converges uniformly to $\infty$ on $E$ (meaning that for every $t>0$, there exists $n \in \mathbb{Z}^+$ such that $f_k(x) > t$ for all integers $k \geq n$ and all $x \in E$).

I also have access to Egorov's Theorem which states the following:

Suppose $(X,\mathcal{S},\mu)$ is a measure space with $\mu(X)<\infty$. Suppose $f_1,f_2,\dots$ is a sequence of $\mathcal{S}$-measurable functions from $X$ to $\mathbb{R}$ that converges pointwise on $X$ to a function $f: X \rightarrow \mathbb{R}$. Then for every $\epsilon>0$, there exists a set $E \in \mathcal{S}$ such that $\mu(X \setminus E) < \epsilon$ and $f_1,f_2,\dots$ converges uniformly to $f$ on $E$.

I think that the above question is just a direct application of the theorem, but that convergence to infinity messes me up a little bit. Would someone be able to help, please?

$\endgroup$

1 Answer 1

1
$\begingroup$

HINT

Assume that $f_n \to +\infty$ pointwise.

Try mimicking Egorov's theorem proof for the sets $$A_{m,n}=\bigcup_{k=n}^{\infty}\{x:f_k(x) \leq m\}$$

https://en.wikipedia.org/wiki/Egorov%27s_theorem Here is a link to the proof,for the case of uniform convergence to a function.Mimic this proof.

Do the same ,when $f_n \to -\infty$ pointwise,using the sets $$A_{m,n}=\bigcup_{k=n}^{\infty}\{x:f_n(x)\geq-m\}$$

$\endgroup$
5
  • $\begingroup$ It's still unclear to me, would you please expand with some more details? $\endgroup$
    – user831321
    Commented Feb 26, 2021 at 22:14
  • $\begingroup$ @hdmovies598 a gave a link to the proof from wiki..try mimicking this proof $\endgroup$ Commented Feb 26, 2021 at 22:16
  • $\begingroup$ Also, can't I just consider a function $f$ that is always evaluated at $\infty$, then $f_k$ would converge pointwise to $f$, and then just apply Egorov's theorem? $\endgroup$
    – user831321
    Commented Feb 26, 2021 at 22:17
  • $\begingroup$ @hdmovies598 you cant use the theorem,because if you see the proof of the theorem you cannot consider the infinity as function. $\endgroup$ Commented Feb 26, 2021 at 22:19
  • $\begingroup$ Ok, what if i consider a function which converges to infinity when approaching $0$, say $f(x) = \frac{1}{x}$? $\endgroup$
    – user831321
    Commented Feb 26, 2021 at 22:21

You must log in to answer this question.