Exercise 1.5.7 (Pointwise convergence in double index)

Question

Let $(X,\mathcal{B},\mu)$ be a measure space, let $\left(fight)_{n\in\mathbb{N}}$ be a sequence of measurable functions from $X$ to $\mathbb{C}$ converging pointwise almost everywhere to a measureble limit $f:X	o\mathbb{C}$ as $n	o\infty$, and for each $n\in\mathbb{N}$ let $\left(f_{n,m}ight)_{n\in\mathbb{N},m\in\mathbb{N}}$ be a sequence of measurable functions from $X$ to $\mathbb{C}$ converging pointwise almost everywhere to $f_n$ as $m	o\infty$.
\begin{enumerate}[label=(oman*)]
\item If $\mu(X)<\infty$, show that there exists a sequence $m_1,m_2,\ldots$ such that $f_{n,m_n}$ converges pointwise almost everywhere to $f$ as $n	o\infty$.
\item Show that the same claim is true if $X$ is $\sigma$-finite.
\end{enumerate}

Elu Thingol · Accepted Answer

\begin{enumerate}[label=( oman*)] \item By Theorem 1.5.9 (Egorov's theorem for finite measure spaces) the conditions of $\left(f ight)_{n\in\mathbb{N}} o f$ and $\left(f_{n,m} ight)_{n\in\mathbb{N},m\in\mathbb{N}}$ pointwise almost everywhere are equivalent to convergence in almost uniform mode. In other words, except for some exceptional set $E_0\in\mathcal{B}$ with $m(E_0)<\delta/2$ for an arbitrary $\delta>0$ we have $$\forall \epsilon > 0 \quad \exists N \in \mathbb{N} \quad \forall n \geq N \quad \forall x\in X/E_0: \quad d(f_n(x),f(x)) \leq \frac{\epsilon}{2}$$ Similarly, for each $n\in\mathbb{N}$ we can find an exceptional set $E_n\in\mathcal{B}$ of measure $\delta/2\cdot 2^n$ such that $$\forall \epsilon > 0 \quad \exists m_n \in \mathbb{N} \quad \forall m \geq m_n \quad \forall x\in X/E_n: \quad d(f_{n,m}(x),f_{n}(x)) \leq \frac{\epsilon}{2}$$ Thus consider a subsequence $\left(f_{n,m_n} ight)_{n\in\mathbb{N}}$ constructed from $\left(f_{n,m} ight)_{n\in\mathbb{N},m\in\mathbb{N}}$ by axiom of choice using above logic. On the exceptional set $E:=\bigcup_{n=0}^{\infty}E_n$ with $\mu(E) \leq \delta/2 + 1/2 \sum_{i=1}^{n} \delta/2^n = \delta$, we then have that $$ ext{For $N_\epsilon\in\mathbb{N}$} \quad \forall n \geq N_\epsilon \quad \forall x \in X/E: \quad d(f_{nm}(x),f(x))\leq d(f_{nm}(x),f_{n}(x)) + d(f_{n}(x), f(x)) \leq \epsilon$$ In other words, $\left(f_{n,m_n} ight)_{n\in\mathbb{N}}$ converges to $f$ almost uniformly, which, by Egorov's theorem, is equivalent to almost everywhere convergence pointwise. \item This follows by applying the above proof countable amount of times to the partition $X = \bigcup_{n=1}^{\infty}X_n$ of $\sigma$-measurable sets $X_n$ with $\mu(X_n)< \infty$ using almost uniform convergence and the $\epsilon/2^n$-trick. \end{enumerate}

Exercise 1.5.7 (Pointwise convergence in double index)

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Exercise 1.5.7 (Pointwise convergence in double index)

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