# Using Egorov's Theorem when the sequence of functions converges pointwise to infinity

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 $$x \in E$$).

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?

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 $$A_{m,n}=\bigcup_{k=n}^{\infty}\{x:f_n(x)\geq-m\}$$

• It's still unclear to me, would you please expand with some more details?
– user831321
Commented Feb 26, 2021 at 22:14
• @hdmovies598 a gave a link to the proof from wiki..try mimicking this proof Commented Feb 26, 2021 at 22:16
• 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?
– user831321
Commented Feb 26, 2021 at 22:17
• @hdmovies598 you cant use the theorem,because if you see the proof of the theorem you cannot consider the infinity as function. Commented Feb 26, 2021 at 22:19
• Ok, what if i consider a function which converges to infinity when approaching $0$, say $f(x) = \frac{1}{x}$?
– user831321
Commented Feb 26, 2021 at 22:21