# Lim sup and liminf with Fatou's lemma

I have some trouble understanding the following homework.

With measure space $$(\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)$$ where $$\mu$$ is the counting measure, we have $$(f_n)_{n\geq1},(g_n)_{n\geq1} \in \mathcal{M}^+$$ as

$$f_n=\mathbb{1}_{\{n\}}$$ and $$g_n= \begin{cases} \mathbb{1}_{\{1\}} \text{ if n is odd}\\ \mathbb{1}_{\{2\}} \text{ if n is even} \end{cases}$$

(i)

Find $$\limsup_{n \rightarrow \infty} f_n, \limsup_{n \rightarrow \infty} g_n$$

(ii)

Show that:

$$\int_\mathbb{N} \limsup_{n \rightarrow \infty} f_n d\mu < \limsup_{n \rightarrow \infty} \int_\mathbb{N} f_n d\mu$$

and

$$\int_\mathbb{N} \limsup_{n \rightarrow \infty} g_n d\mu > \limsup_{n \rightarrow \infty} \int_\mathbb{N} g_n d\mu$$

I have:

$$\limsup_{n \rightarrow \infty} f_n=0, \limsup_{n \rightarrow \infty} g_n=0$$

This satisfies the first inequality in (2) as the integral is always 1 so 0<1. However the same approach goes complete wrong in the second inequality where I have 0>2 which is absurd.

Can anyone see what is wrong? I think maybe I am misunderstanding the limsup or liminf concepts :( I am assuming limsup and liminf are pointwise

You've made two mistakes in dealing with $$g_n.$$
For the first, note that $$g_n(1) = \begin{cases}1 & \text{ if } n \text{ is odd} \\ 0 & \text{ otherwise}\end{cases}$$ so that $$\left(\limsup g_n\right)(1) = 1$$
Doing the same [point-wise] analysis with $$g_n(2),$$ we see that $$(\limsup g_n)(2) = 1$$ and it will be zero for all other inputs. That is, $$\limsup g_n = 1_{\{1,2\}}.$$ Use this to evaluate $$\int_\mathbb{N} \limsup g_n\,d\mu.$$
For the second, you should re-visit $$\int_\mathbb{N} g_n\,d\mu.$$ It's never going to be equal to $$2.$$
• Ok, as I understand then $\lim_\mathbb{N} \limsup g_n d\mu=2 > 1=\limsup \int_\mathbb{N} g_n d\mu$. But isn't x chosen first and then limsup is taken afterwards, so it becoms pointwise? – Daniel Sep 22 '19 at 8:25