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Suppose that we have $(\Omega,\mu)$ a $\sigma$-finite measure space. I have the following question. (Assume that $(f_i)_{i\in I}$ be an increasing net of positive measurable functions such that $f_i\uparrow f$ pointwise. Suppose we know that $f$ is measurable. If $\sup_{i\in I}\int f_i$ is finite, can we say that $\int f$ is finite and $\lim_{i}\int f_i=\int f$? If not can it hold under the stronger assumption that $\int f$ is finite?

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    $\begingroup$ The Dieudonné measure on $\omega_1$ is a counterexample. Or, Lebesgue measure on $[0,1]$ under the continuum hypothesis CH: you may then write the constant function 1 as the pointwise limit of an increasing net of functions with countable support. $\endgroup$ May 24, 2022 at 3:49
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    $\begingroup$ Take $I$ to be the set of finite subsets of $\Omega$, and assume that all singletons are measurable. The net $(\chi_{S})_{S \in I}$ is monotone and converges to the constant $1$ function pointwise (just follow the definition). So for any non-zero measure vanishing on singletons we have a counterexample. $\endgroup$ May 24, 2022 at 7:12
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    $\begingroup$ If $\Omega$ is a topological space, all open sets are measurable, and $f$ and all $f_i$ are lower semicontinuous, and you have a $\tau$-additive measure $\mu$, then this does hold. Radon measures, such as Lebesgue measure, are $\tau$-additive. The Dieudonné measure mentioned by Nate Eldredge is a classical non-$\tau$-additive measure. $\endgroup$ May 24, 2022 at 7:16
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    $\begingroup$ @Abeginnermathmatician Sorry, I didn't see that your measure was $\sigma$-finite and not finite. Let $Y$ be a measurable set with $0 < \mu(Y) < \infty$ (which exists for any non-zero $\sigma$-finite measure). Take $I$ to be the finite subsets of $Y$. The net $(\chi_S)_{S \in I}$ is monotone and converges pointwise to $\chi_Y$. So if all singletons in $Y$ are measurable with measure zero this is a counterexample. $\endgroup$ May 24, 2022 at 13:34
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    $\begingroup$ @NateEldredge: where is the continuum hypothesis used in the Lebesgue measure example? $\endgroup$ May 24, 2022 at 13:40

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