Exercise 1.5.1 (Linearity of convergence)

Question

Let $(X,\mathcal{B},\mu)$ be a measure space, let $\left(fight)_{n\in\mathbb{N}},\left(gight)_{n\in\mathbb{N}}$ be sequences of measurable functions from $X$ to $\mathbb{C}$, and let $f,g:X	o\mathbb{C}$ be another measurable functions. Show that
\begin{enumerate}[label=(oman*)]
	\item If $\left(fight)_{n\in\mathbb{N}}$ converges to $f$ along one of the above seven modes of convergence if and only if $|f_n-f|$ converges to 0 along the same mode.
	\item If $\left(fight)_{n\in\mathbb{N}}$ converges to $f$ along one of the above seven modes of convergence, and $\left(gight)_{n\in\mathbb{N}}$ converges to $g$ along the same mode, then for any $c \in \mathbb{C}$, $(cf_n+g)_{n\in\mathbb{N}}$ converges to $cf+g$ along the same mode.
	\item (Squeeze test) If $\left(fight)_{n\in\mathbb{N}}$ converges to $0$ along one of the above seven modes of convergence, and $g_n\leq f_n$ pointwise for each $n\in\mathbb{N}$, show that $\left(gight)_{n\in\mathbb{N}}$ converges to 0 along the same mode.
\end{enumerate}

Elu Thingol · Accepted Answer

$
ewcommand{\intx}[1]{\int_{X} #1 \mathop{}\!\mathrm{d}{\mu}}$
\begin{enumerate}[label=(oman*)]
	\item For convergence modes (i)-(v) this is trivial since $|f_n-f| = \left| |f_n-f| - 0 ight|$. Similary, for (vi) we have $\intx{|f_n-f|} = \intx{\left| |f_n-f| - 0 ight|}$. Finally, $\{x\in X:  |f_n-f| \geq \epsilon \} = \{x\in X: \left| |f_n-f| - 0 ight|\geq \epsilon \}$, and so we also have convergence in measure.
	\item For convergence modes (i)-(v) we use the triangle property of the underlying metric, and see that
      $$\left| (cf_n+g_n) - (cf+g) ight| \leq |c||f_n-f| + |g_n-g| $$
      For (iv), we combine the above property with the triangle inequality for integrals
      $$\intx{\left| (cf_n+g_n) - (cf+g) ight|} \leq |c|\intx{|f_n-f|} + \intx{|g_n-g|} $$
      Finally for the convergence in measure we use the additivity of the measure combined with
      \begin{align*}
      \left\{x \in X: \left| (cf_n+g_n) - (cf+g) ight| \geq \epsilon ight\}
      &\subseteq \left\{x \in X: |c||f_n-f| + |g_n-g| \geq \epsilon ight\}\
      &\subseteq \left\{x \in X: |c||f_n-f| \geq \epsilon ight\} \cup \left\{x \in X: |g_n-g| \geq \epsilon ight\}
      \end{align*}
	\item Again, the statement is trivial for convergence modes (i)-(v) since $|g_n|\leq f_n \implies |g_n| \leq |f_n|$. $L^1$ convergence follows the same way by monotonicity of integrals, and the convergence in measure follows from monotonicity of measure and $\{x \in X: |f_n| \geq \epsilon\} \supseteq \{x \in X: |g_n| \geq \epsilon\}$.
\end{enumerate}

Exercise 1.5.1 (Linearity of convergence)

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Exercise 1.5.1 (Linearity of convergence)

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