Let ((X, mathscr{A}, mu)) be a (sigma)-finite measure space, (f in mathcal{M}(mathscr{A})) and (1 leqslant p (i)

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Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space, \(f \in \mathcal{M}(\mathscr{A})\) and \(1 \leqslant p

(i) If \(f \in L^{p}(\mu)\), then \(\|f\|_{p}=\sup \left\{\int f g d \mu: g \in L^{q}(\mu),\|g\|_{q} \leqslant 1ight\}\).

(ii) Show that we can replace in (i) the set \(L^{q}(\mu)\) with a dense subset \(\mathcal{D} \subset L^{q}(\mu)\).

(iii) If \(f g \in L^{1}(\mu)\) for all \(g \in L^{q}(\mu)\), then \(f \in L^{p}(\mu)\).

[consider \(g \mapsto I_{f}(g)=\int|f| g d \mu\) and use Theorem 21.5 .]

Data from theorem 21.5

Theorem 21.5 (Riesz) Let (X,,u) be a o-finite measure space, p = [1,00) and qe (1,00] conjugate indices.

This means that v < < is absolutely continuous with respect to , and the Radon Nikodm theorem (Theorem 20.2)

Step 5. Assume that p(X)=o. Let (A) MEN be a sequence in with A, 1X and (4) <0. As in the proof of the

Therefore, J(u):=1(h/Pu) is a positive linear functional on LP(), and the first four steps show that there is

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