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Nov 21, 2022 at 15:58 comment added Michael Greinecker I guess somlike the following should work: Take the positive and negative parts of $f$ and do the problem for both separately using monotone approximations by simple functions. Then everything works by the monotone convergence theorem and you just need to show that their averages are finite.
Nov 21, 2022 at 15:01 history edited Akira CC BY-SA 4.0
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Nov 21, 2022 at 15:00 comment added Akira @MichaelGreinecker It's a typo. I meant $\int f \mathrm d \mu_y$, not $\int F \mathrm d \mu_y$.
Nov 21, 2022 at 14:59 comment added Michael Greinecker What is $F$ here?
Nov 21, 2022 at 12:57 history edited Akira CC BY-SA 4.0
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Nov 21, 2022 at 12:51 history edited Akira CC BY-SA 4.0
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Nov 21, 2022 at 12:45 history asked Akira CC BY-SA 4.0