(adsbygoogle = window.adsbygoogle || []).push({}); The limit of an "almost uniformly Cauchy" sequence of measurable functions

I'm trying to understand the proof of theorem 2.4.3 in Friedman. I don't understand why f must be measurable. The "first part" of the corollary he's referring to says nothing more than that a pointwise limit of a sequence of measurable functions is measurable. But it seems to me that f is just an almost uniform limit, not a pointwise limit. Almost uniform convergence implies convergence a.e, but not (everywhere) pointwise convergence.

This is my (possibly incorrect) interpretation of what we're doing. We find that there's a subsequence that's "almost uniformly Cauchy" in the sense I'm about to describe, so now the goal is to use that sequence to define a new function, and then we prove that it's measurable. Let's try this for an arbitrary sequence ##\langle g_n\rangle## of measurable functions that 's "almost uniformly Cauchy" in the sense that for all ##\varepsilon>0##, there's a measurable set ##E## such that ##\mu(E)<\varepsilon## and ##\langle g_n(x)\rangle## is uniformly Cauchy on ##E^c##. What I mean by that is that there's an ##N\in\mathbb Z^+## such that for all ##n,m\in\mathbb Z^+## and all ##x\in E^c##,

$$n,m\geq N\ \Rightarrow\ |g_n(x)-g_m(x)|<\varepsilon.$$ The assumption that ##\langle g_n\rangle## is almost uniformly Cauchy implies that for all ##k\in\mathbb Z^+##, there's a measurable set ##E_k## such that ##\mu(E_k)<\frac{1}{k}## and ##\langle g_n(x)\rangle## is uniformly Cauchy on ##E^c##. This allows us to define a function ##g:X\to\mathbb R## by

$$

g(x)=

\begin{cases}

\lim_n g_n(x) &\text{ if }x\in\bigcup_{k=1}^\infty E_k{}^c\\

0 &\text{ if }x\notin\bigcup_{k=1}^\infty E_k{}^c.

\end{cases}

$$

(How else can we define g?) Then we're supposed to be able to show that this g is measurable. Is this what Friedman has in mind? Since

$$\mu\bigg(\bigcap_{k=1}^\infty E_k\bigg) \leq\mu(E_k)<\frac{1}{k}$$ for all k, we have almost uniform convergence, but not (everywhere) pointwise convergence.

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# The limit of an almost uniformly Cauchy sequence of measurable functions

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