Extension by continuity. Let \(T: L^{2}(\mu) ightarrow L^{2}(\mu)\) be a linear operator such that \(\|T u\|_{p} \leqslant\) \(c\|u\|_{p}\) for \(u \in L^{2}(\mu) \cap L^{p}(\mu)\) for some \(p eq 2\). Show that there is a unique extension \(\tilde{T}: L^{p}(\mu) ightarrow L^{p}(\mu)\) defined by \(L^{p}-\lim _{n ightarrow \infty} T u_{n}\) for any sequence \(u_{n} \in L^{2}(\mu) \cap L^{p}(\mu)\) with \(u_{n} ightarrow u\) in \(L^{p}(\mu)\). If \(T\) is monotone, i.e. \(u \leqslant w \Longrightarrow T u \leqslant T w\), then \(u_{n} \uparrow u\) a.e. implies that \(T u_{n} \uparrow T u\) a.e.

[ have a look at the remark preceding Theorem 27.5 .]

Data from theorem 27.5

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Theorem 27.5 Let (X, A, ) be a o-finite measure space. The conditional expec- tation E has an extension mapping LP (A) to LP(G) for any p= [1, ]. Proof Using the method of extension by continuity, it is enough to check that Eup < cup for all p and u L(A)nIP(A). Step 1. p=o. If u L(A) L (A) we get from Theorem 27.4 and the observation that u\/||u|| <1 E (u/|u)|