Exercise 1.5.4

Question

Let $(X,\mathcal{B},\mu)$ be a measure space with $\mu(X)<\infty$, let $\left(fight)_{n\in\mathbb{N}}$ be a sequence of measurable functions from $X$ to $\mathbb{C}$, and let $f:X	o\mathbb{C}$ be another measurable function. Show that if $\left(fight)_{n\in\mathbb{N}}$ converges to $f$ in $L^\infty$-norm, then $\left(fight)_{n\in\mathbb{N}}$ also converges to $f$ in $L^1$-norm.

Elu Thingol · Accepted Answer

$
ewcommand{\intx}[1]{\int_{X} #1 \mathop{}\!\mathrm{d}{\mu}}$
By definition, we have
$$\forall \epsilon>0 \quad \exists N\in\mathbb{N} \quad \forall n \geq N \quad \forall x \in X/E: \quad |f(x) - f_n(x)| \leq \frac{\epsilon}{\mu(X)}$$
for some null set $E\in \mathcal{B}$. We then have
$$\intx{|f-f_n|} = \int_{E} |f-f_n| \mathrm{d}\mu + \int_{X/E} |f-f_n|\mathrm{d}\mu \leq \int_{X/E} \frac{\epsilon}{\mu(X)} \leq \mu(X/E)\cdot \frac{\epsilon}{\mu(X)} \leq \epsilon.$$
Thus, $\intx{|f-f_n|}$ can be reduced arbitrarily and consistently after some $N\in\mathbb{N}$, and so $\left(fight)_{n\in\mathbb{N}}$ converges to $f$ in $L^1$-norm.

Exercise 1.5.4

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Exercise 1.5.4

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