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While self studying Lebesgue integration from Tom M Apostol I am unable to think about an how to deduce a line in the proof whose image I am adding( line is highlited.

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I am not able to deduce the line that the sequence { $G_n(x) $ } decreases almost everywhere and hence converges everywhere.

I am unable to get how the author wrote that {$ G_n (x) $ } is decreasing? and how that implies that it's convergent.

Kindly help.

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  • $\begingroup$ The fact $G_n(x) $ is decreasing is already explained in "since $A\subseteq B$ implies $\sup A \leq \sup B$". $\endgroup$
    – Paramanand Singh
    Commented Jun 19, 2020 at 6:29
  • $\begingroup$ The convergence of $G_n(x) $ is explained afterwards by showing that if $f_n(x) \to f(x) $ then $G_n(x) \to f(x) $. The details are in next page of the book. $\endgroup$
    – Paramanand Singh
    Commented Jun 19, 2020 at 6:33

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$\{G_n (x)\}$ is decreasing because it's defined as supremum of $\{f_n(x), f_{n+1}(x),...\}$. $G_1 (x)=\sup\{f_1(x),f_2(x),f_3(x),...\}$ and $G_2(x)=\sup\{f_2(x), f_3(x),...\}$, so, depending on $f_1(x)$, $G_2(x)\leq G_1(x)$ and so on. And this is why $\{G_n (x)\}$ converges a.e.: $\{f_n(x)\}$ converges a.e. and for fixed $x_0$, such that number sequence $\{f_n(x_0)\}$ converges, we have $$\lim_{n\to \infty}f_n(x_0) = \lim_{n\to \infty}\left( \sup_{m\geq n}f_n(x_0)\right) = \lim_{n\to \infty}\left( \inf_{m\geq n}f_n(x_0)\right)$$ So, $$\lim_{n\to \infty}f_n(x_0) = \lim_{n\to \infty}G_n(x_0)$$ hence $\{G_n (x)\}$ converges in points, where $\{f_n (x)\}$ also converges.

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