analysis-integration measure Given a measurable space $(X, V, m)$ and  $\{F_{n}\}_{1}^{\infty}\subset $ $V $  is a sequence of sets such that $m(F_{n})\leq$ $e^{-n}$  $\forall {n}.$ show that the functions $h(x)=\sum_{1}^{\infty} {
\chi_E{_n}(x)}$ and $g(x)=\sum_{1}^{\infty} {n^{t}\chi_E{_n}(x)}$ belongs to $L^p $ for all   $ 0<p, t<\infty$.
 A: Since 
$$
\left( \int_X \left( \sum_{n=1}^N  \chi_{F_n} \right)^p dm \right)^{1/p} \le \sum_{n=1}^N \left( \int_X \chi_{F_n} dm \right)^{1/p}
$$
(i.e. $\|f+g\|_p \le \|f\|_p + \|g\|_p$ and by induction, the same goes for $n$ functions), by letting $N$ go to infinity, we can use the monotone convergence theorem because since $0 \le \sum_{n=1}^N \chi_{F_n} \le \sum_{n=1}^{N+1} \chi_{F_n}$, this sequence is increasing, thus
$$
\| h \|_p = \left\| \lim_{N \to \infty} \sum_{n=1}^N \chi_{F_n} \right\|_p = \lim_{N \to \infty} \left\| \sum_{n=1}^N \chi_{F_n} \right\|_p \le \lim_{N\to \infty} \sum_{n=1}^N \| \chi_{F_n} \|_p = \sum_{n=1}^{\infty} \|\chi_{F_n} \|_p
$$
(for the second equality, putting both sides at $p^{\text{th}}$ power is like looking at integrals, thus I can use monotone convergence). Now using the properties of the $F_n$'s we obtain
$$
\begin{align*}
\|h\|_p & \le \sum_{n=1}^{\infty} \left( \int_X \chi_{F_n} \, dm \right)^{1/p} 
= \sum_{n=1}^{\infty} \left( m(F_n) \right)^{1/p} \\
& \le \sum_{n=1}^{\infty} (e^{-n})^{1/p} = \sum_{n=1}^{\infty} (e^{-1/p})^n = \frac{e^{-1/p}}{1-e^{-1/p}} < \infty.
\end{align*}
$$
(I assumed your $E_n$'s were the $F_n$'s.)
Some similar argument should hold for the next one, I just gave you the ideas so that you can still try the next. If you want me to work it out the second one I will, just comment.
Hope that helps,
EDIT : You seem to have trouble with the second case. Notice that the same steps go for $g_t$ up to some point, i.e.
$$
\|g_t \|_p \le \sum_{n=1}^{\infty} n^t (e^{-1/p})^n.
$$
Now the series $\sum_{n=1}^{\infty} n^t x^n$ converges when $|x|<1$ for all $t \in \mathbb R$. If $t \le 0$ this is trivial (the sum is bounded by the geometric series, thus converges), so let me suppose that $t > 0$. If $t$ is not an integer, then the sum computed at $t$ is less than the sum computed at an integer greater than $t$, so I'll assume $t$ is an integer. Then
$$
\sum_{n=1}^{\infty} n^t |x|^n \le \sum_{n=1}^{\infty} (n+1)(n+2)\cdots(n+t)|x|^n
$$
and the sum on the right is the $t^{\text{th}}$ derivative of the geometric series, thus it converges absolutely with the same radius of convergence.
A: Assuming $E^n = F_n$.
To show $h \in L^p$ you need to show $(\int_X |h|^p )^{1/p} < \infty$:
$$ \left( \int_X |h|^p \right)^{1/p} = \left( \int_X |\sum \chi_{F_n}|^p \right)^{1/p} \leq \left( \int_X \left( \sum |\chi_{F_n}| \right)^p \right)^{1/p}$$
Then using the Lebesgue dominated convergence theorem you know that you can do this:
$$ \left( \int_X \left( \sum |\chi_{F_n}| \right)^p \right)^{1/p} = \left( \int_X \lim_{N \rightarrow \infty} \left( \sum_{n=1}^N |\chi_{F_n}| \right)^p \right)^{1/p} = \left(\lim_{N \rightarrow \infty} \int_X \left(\sum_{n=1}^N |\chi_{F_n}| \right)^p \right)^{1/p}$$
i.e. you can swap limit and integral and then because you have a finite sum you can swap integral and sum. Let's put things together: 
$$ \begin{align*}
\left( \int_X |h|^p \right)^{1/p} = \left( \int_X \Big |\sum_{n=1}^\infty \chi_{F_n} \Big |^p dm \right)^{1/p} \\

\stackrel{\Delta-ineq.}{\leq} \left( \int_X \left( \sum_{n=1}^\infty  |\chi_{F_n}| \right)^p dm \right)^{1/p} \\

= \left( \int_X \left( \lim_{N \rightarrow \infty} \sum_{n=1}^N |\chi_{F_n}| \right)^p dm \right)^{1/p} \\

\stackrel{cont. of ()^p}{=} \left( \int_X \lim_{N \rightarrow \infty} \left( \sum_{n=1}^N |\chi_{F_n}| \right)^p dm \right)^{1/p} \\

\stackrel{Lebesgue}{=} \left( \lim_{N \rightarrow \infty} \int_X \left( \sum_{n=1}^N |\chi_{F_n}| \right)^p dm \right)^{1/p} \\

\stackrel{\|.\| \Delta ineq.}{\leq} \lim_{N \rightarrow \infty} \sum_{n=1}^N \left( \int_X |\chi_{F_n}|^p  dm \right)^{1/p} \\

\stackrel{\chi \in \{0,1\}}{=}\lim_{N \rightarrow \infty} \sum_{n=1}^N \left( \int_X \chi_{F_n}  dm \right)^{1/p} \\

=  \lim_{N \rightarrow \infty} \sum_n^N \left( m(F_n) \right)^{1/p} \\

\leq   \lim_{N \rightarrow \infty} \sum_n^N \left( e^{-n} \right)^{1/p} < \infty
\end{align*}$$
Where the last inequality comes from the fact that this is a geometric series with $\frac{1}{e^{\frac{n}{p}}} < 1$ for $n$ large enough so it converges.
Edit
For the second function note that there exists an $N_0$ such that for $n > N_0: \Big | \frac{n^t}{e^n} \Big | < 1$. Then $$ \lim_{N \rightarrow \infty} \sum_{n=1}^N \frac{n^t}{e^n} = \sum_{n=1}^{N_0} \frac{n^t}{e^n} + \lim_{N \rightarrow \infty} \sum_{n={N_0}}^N \frac{n^t}{e^n} $$
