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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.

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    @ t.d, I wanted to make use of the definition of L^p, that was why I deleted the other equations.2011-11-12

2 Answers 2

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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.

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    Use the fact that \sum_{n=1}^{\infty} n^t x^n < \infty when |x|< 1 and $t \in \mathbb R$.2011-11-12
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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} $

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    @PatrickDaSilva: thanks, I had corrected some other typos earlier, thought I had gotten them all : S2011-11-13