Let ((Omega, mathscr{A}, mathbb{P})) be a probability space. Adapt the proof of Theorem 22.9 to show that

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Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. Adapt the proof of Theorem 22.9 to show that a sequence \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{L}^{1}(\mu)\) is uniformly integrable if it is bounded in some space \(\mathcal{L}^{p}(\mathbb{P})\) with \(p>1\), i.e. if \(\sup _{n \in \mathbb{N}}\left\|u_{n}ight\|_{p}

Use Vitali's convergence theorem to construct an example illustrating that \(\mathcal{L}^{1}\) boundedness of \(\left(u_{n}ight)_{n \in \mathbb{N}}\) does not guarantee uniform integrability.

Data from theorem 22.9

Theorem 22.9 Let (X, A, ) be some measure space and FCL(A). Then the following statements (i)-(v) are

If (X, A, p) is a o-finite measure space, (i) (v) are also equivalent to sup /\u\ < 0; (vi) (a) sup

(b) lim sup [ [uldu = 0 for all decreasing sequences (An)nEN CA, An 40. 11-700 UEJ J An [Note: it is not

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