Let \((X, \mathscr{A}, \mu)\) be a measure space. The space \(\mathcal{L}^{p}(\mu)\) is called separable if there exists a countable dense subset \(\mathscr{D}_{p} \subset \mathcal{L}^{p}(\mu)\). Show that \(\mathcal{L}^{p}(\mu)\), \(p \in(1, \infty)\), is separable if, and only if, \(\mathcal{L}^{1}(\mu)\) is separable.

[ use Riesz's convergence theorem, Theorem 13.10.]

Data from theorem 13.10

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Theorem 13.10 (F. Riesz) Let (Un)neN CLP (), p= [1, ), be a sequence such that limn un(x) = u(x) for almost every xEX and some u LP (). Then lim unu||p=0 11700 lim ||un|p|u||p. 110 (13.11) Proof The direction in (13.11) follows from the lower triangle inequality4 |||un|pu||p| < ||un-up for ||- ||p. For we observe that un-up < (|un| + |u) <2 max{un, up} <2 (\uP+uP), and we can apply Fatou's lemma (Theorem 9.11) to the sequence 2P (un+uP)-un-up > 0 4 In the same way as ||| -|b||