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(Levi) If $\{f_n\}$ is a sequence in $L(I)$ such that (i) $f_n$ increases a.e. on $I$ (ii) $\lim_{n\to\infty}\int_I f_n$ exists then $\{f_n\}$ converges a.e. to a function $f\in L(I)$ and $$\int_I f=\lim_{n\to\infty}\int_I f_n.$$ :: $[10.22]S(I)\to U(I)$, $[10.23]U(I)\to U(I)$, $[10.24]L(I)\to L(I)$, $[10.25]$ series. <hr> (Dominance Theorem 10.27) If $\{f_n\}$ is a sequence in $L(I)$ such that (i) $\{f_n\}$ converges a.e. to a limit function $f$. (ii) (Dominated condition) Exists nonnegative $g\in L(I)$ such that $$|f_n(x)|\leq g(x)\text{ a.e. on }I.$$ then $f\in L(I)$, $\{\int_I f_n\}$ converges and $$\int_I f=\lim_{n\to\infty}\int_I f_n.$$
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Sun, 17 Jan 2021 03:09 GMT