Exercise 1.5.16 (Monotone convergence theorem)

Question

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ewcommand{\intx}[1]{\int_{X} #1 \mathop{}\!\mathrm{d}{\mu}}$
Let $\left(fight)_{n\in\mathbb{N}}:X	o [0,+\infty)$ be a sequence of measurable, monotone functions that are non-decreasing in $n\in\mathbb{N}$ and such that $\sup_{n \in \mathbb{N}} \intx{f_n}< \infty$. Show that $\left(fight)_{n\in\mathbb{N}}$ converges in $L^1$ norm to $\sup_{n\in\mathbb{N}}f_n$.

Elu Thingol · Accepted Answer

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ewcommand{\intx}[1]{\int_{X} #1 \mathop{}\!\mathrm{d}{\mu}}$
By the original monotone convergence theorem (Theorem 1.4.43)
$$0 = \intx{\sup_{k \in \mathbb{N}} f_k} - \sup_{n \in \mathbb{N}} \intx{f_n} = \sup_{n \in \mathbb{N}} \intx{\left(\sup_{k \in \mathbb{N}} f_k - f_night)} = \sup_{n \in \mathbb{N}} \intx{\left|\sup_{k \in \mathbb{N}} f_k - f_night|}.$$

Exercise 1.5.16 (Monotone convergence theorem)

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Exercise 1.5.16 (Monotone convergence theorem)

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