Exercise 1.5.18 (Egoroff's theorem IV)

Question

Let $\left(fight)_{n\in\mathbb{N}}:X	o\mathbb{C}$ be a dominated sequence of measurable functions, and let $f:X	o\mathbb{C}$ be another measurable function. Show that $\left(fight)_{n\in\mathbb{N}}$ converges pointwise almost everywhere to $f$ if and only if $\left(fight)_{n\in\mathbb{N}}$ converges almost uniformly to $f$.

Elu Thingol · Accepted Answer

$
ewcommand{\intx}[1]{\int_{X} #1 \mathop{}\!\mathrm{d}{\mu}}$
As always, throughout the proof we safely discard the "almost everywhere" part of the pointwise convergence.
    
If $\left(fight)_{n\in\mathbb{N}}$ converges almost uniformly to $f$, then the a.e. pointwise convergence follows by \href{./exercise_1-5-2}{Exercise 1.5.2} (iv).

The other direction is practically  \href{./exercise_1-4-31}{Exercise 1.4.31} but with a slight change: we don't have the finite measure assumption anymore, but in its place we assumed that there is an absolutely integrable $g\in L^1(X)$ which dominates our sequence $\forall n\in\mathbb{N}: f_n \leq g$ almost everywhere. It is easy to verify that this is enough to make sure that the sets $E_{N,m}$ are still finite in measure:
\begin{align*}
      \forall \frac{1}{m}>0: \quad E_{N,m} &= \left\{x \in X/E_0	ext{ $\big|$ } \exists n\geq N: 	ext{ } \left|f_n(x)-f(x)ight|\geq\frac{1}{m}ight\}\
                                           &\subseteq \left\{x \in X/E_0	ext{ $\big|$ } \left|f_n(x)-f(x)ight|\geq\frac{1}{m}ight\}\
                                           &\subseteq \left\{x \in X/E_0	ext{ $\big|$ } 2g(x) \geq\frac{1}{m}ight\} \leq m \cdot \intx{2g} < \infty
\end{align*}
At this point we can continue the proof of \href{./exercise_1-4-31}{Exercise 1.4.31}.

Exercise 1.5.18 (Egoroff's theorem IV)

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Exercise 1.5.18 (Egoroff's theorem IV)

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