Let (left(u_{n}ight)_{n in mathbb{N}}) be a sequence of measurable functions on a (sigma)-finite measure space ((X, mathscr{A},

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Let \(\left(u_{n}ight)_{n \in \mathbb{N}}\) be a sequence of measurable functions on a \(\sigma\)-finite measure space \((X, \mathscr{A}, \mu)\) and assume that \(u_{n} \stackrel{\mu}{ightarrow} u\).

(i) Show that \(\lim _{n ightarrow \infty} \int\left|u_{n}-uight| \wedge \mathbb{1}_{A} d \mu=0\) for every \(A \in \mathscr{A}\) with finite measure \(\mu(A)

(ii) Show that every subsequence \(\left(u_{n}^{\prime}ight)_{n \in \mathbb{N}} \subset\left(u_{n}ight)_{n \in \mathbb{N}}\) contains a further subsequence \(\left(u_{n}^{\prime \prime}ight)_{n \in \mathbb{N}} \subset\left(u_{n}^{\prime}ight)_{n \in \mathbb{N}}\) such that \(\lim _{n ightarrow \infty} u_{n}^{\prime \prime}=u\) almost everywhere.

[ apply (i) with \(A=A_{i}\) from a sequence \(A_{i} \uparrow X, \mu\left(A_{i}ight)

(iii) Use (ii) to give an alternative proof of Lemma 22.5 .

Data from corollary 13.8

Corollary 13.8 Let (Un)nEN CLP (u), pe [1, ) with LP-limn Un=u. Then there exists a subsequence (un(k))keN

Data from lemma 22.5

Lemma 22.5 Let (Un)neNCM(A) be a sequence with unu. If\u, un

For >0 and R> 0 we find [ 124 1 dps = [1 d =  {\makse)! \uan/P d + [{\m[> R} n{w e} E})=0. 11+00 Now we can

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