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

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Theorem 21.5 (Riesz) Let (X,,u) be a o-finite measure space, p = [1,00) and qe (1,00] conjugate indices. Every positive linear functional I: IP () R is of the form (21.3), i.e. there is some unique feL(), f20, such that I = I Proof Step 1. Uniqueness. If If=Ig, then If-g=0. Define h:=f-g and observe that (h|9-P = |h|(9-1ph L (u). Thus, sgn (h) h LP (u), and we get 0=1k(sgn(h)|h|4-1) = { \h\ d h=0 -a.e. =g-a.e. Step 2. Existence if (X)