Sequence of measures in L-p Spaces

Question

Let $(X,\mathscr{A},\mu)$ be a measure space and $\{E_n\}_{n \in \mathbb{N}} \subset \mathscr{A}$ such that $$X=\bigcup_{n=1}^{\infty}E_n \quad \text{and} \quad E_n \subset E_{n+1} \quad(\forall n\in \mathbb{N}) $$ $(i)$ Let $f \in L^1(X,\mathscr{A},\mu).$ Show that $$\int_{E_n^{c}}|f|\;d\mu \longrightarrow 0 \quad \text{as} \quad n \longrightarrow \infty.$$ $(ii)$.Let $g \in L^1(\mathbb{R}).$ Show that $$\lim_{n \to \infty}\int_{|x|>n}|g(x)|dx=0.$$

I know that i should define a measure, say $\psi: \mathscr{A} \longrightarrow [0,\infty)$ by $$\psi(E)=\int_E |f|\;d\mu \quad (\forall E \in \mathscr{A}).$$From here i don't know what to do. Any help or hint would be much appreciated!

Answer 1

In both cases, one can use the monotone convergence theorem, first with $f_n\colon x\mapsto \left\lvert f(x)\right\rvert\mathbf 1_{E_n}(x)$ and for the second question with $g_n\colon x\mapsto \left\lvert g(x)\right\rvert\mathbf 1_{[-n,n]}(g(x))$.

Answer 2

$\textbf{Problem 1.}$ $$\int_{{E_n}^c}|f|d\mu=\int |f| d\mu - \int _{E_n}fd\mu =\int |f|(1-\chi_{E_n})d\mu$$

where $\chi$ is a characteristic function. Noth that $|f|(1-\chi_{E_n})\leq|f|$ and $|f|$ is in $L^1(\mu)$.

By Lebesgue's Dominatied Convergence Theorem, $$\lim_{n\rightarrow\infty}\int_{{E_n}^c}|f|d\mu=\lim_{n\rightarrow\infty}\int |f|(1-\chi_{E_n})d\mu=\int \lim_{n\rightarrow \infty}|f|(1-\chi_{E_n})d\mu $$

$$\int|f|(1-\chi_{\bigcup_{n=1}^\infty En})d\mu=\int |f|(1-\chi_X)d\mu=0$$

$\textbf{Problem 2.}$

Likewise previous problem, you can use Lebesgue's Dominated Convergence Theorem.

$\textbf{Hint :}$ $\displaystyle\int_{|x|>n} |g(x)|dx = \int_{-\infty} ^{\infty}|g(x)|\chi_{{E_n}}dx $ and $|g(x)|\chi_{E_n} \leq |g(x)|$, $g\in L^1(\mathbb{R})$ where $E_n=\left\lbrace x \in \mathbb{R} : |x|>n \right\rbrace$, use LDCT