Let ( u n ) n N ( u n ) n N be a

Let ${\left(u_n\right)}_{n\in N}$ be a sequence of integrable functions on $(X,A,\mu )$. Show that, if ${\sum }_{n=1}^{\infty }\int \left|u_n\right|d\mu \infty$

$$\int \sum _{n=1}^{\infty }{u}_{n}d\mu =\sum _{n=1}^{\infty }\int u_{n}d\mu$$

[use Corollary 9.9 to see that the series ${\sum }_n{u}_n$ converges absolutely for almost all $x\in X$. The rest is then dominated convergence.]

Data from corollary 9.9

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Let (un)neN CM (A). Then 1 un is measurable and we have undu= |undu | n=] (including the possibility too = +00). Proof Set sm := u] + u2 + ... h=1 (9.6) + um and use Property 9.8(iii) and Theorem 9.6.