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
Best Answer
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 $2.$