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Point (a) of Exercise 5 Chapter 1 in Rudin's "Real and complex analysis" requires to prove the following:

If $f,g:X \to [-\infty, +\infty]$ are measurable then $\{x:f(x)<g(x)\}$ and $\{x:f(x)=g(x)\}$ are measurable.

Now the natural approach (and the one that you can find pretty much everywhere the exercise has been discussed) is to notice that $f-g:X \to [-\infty,+\infty]$ is measurable and then to write both sets as preimages of Borel subsets of $[-\infty, +\infty]$. However, Rudin has never proved that the sum of extended-real measurable functions is measurable (he has only proved it for non-negative extended-real functions). Now, I have found a proof of this fact by elementary means (see for instance here: https://www.cmi.ac.in/~prateek/measure_theory/2010-09-24.pdf), however that raises the question of why hasn't Rudin followed this more general road and instead only dealt, in a more obscure way, with a particular case in section 1.22. Is there something I am missing about his approach that makes it favourable?

Matteo Menghini
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  • The general case might follow from the nonnegative case, explaining Rudin's omission. Or Rudin just made a genuine oversight by not demonstrating the general result. – FShrike Aug 13 '23 at 11:02
  • @FShrike You are correct about the fact that the general case follows from the nonnegative case, but just for addition. Indeed multiplication does not follow as trivially. – Matteo Menghini Aug 13 '23 at 11:10

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It is not an oversight - the problem is that there is no guarantee $g-f$ of two extended-real measurable functions is even well-defined. This happens if eg. $f(x) =g(x) = \infty$.

The correct solution to the exercise is instead to look at a case-by-case basis, in the 9 different cases: \begin{align} \Omega_1 &:= \{x: f(x) = \infty, g(x) = \infty\},\\ \Omega_2 &:=\{x: f(x) = \infty, g(x) = \in \mathbb{R}\},\\ \Omega_3 &:= \{x: f(x) = \infty, g(x) = -\infty\},\\ \Omega_4 &:= \{x: f(x) \in \mathbb{R}, g(x)= \infty\},\\ \Omega_5 &:= \{x: f(x) \in \mathbb{R}, g(x)\in \mathbb{R}\},\\ \Omega_6 &:= \{x: f(x) \in \mathbb{R}, g(x)=-\infty\},\\ \Omega_7 &:= \{x: f(x) = -\infty, g(x)\in =\infty\},\\ \Omega_8 &:= \{x: f(x) = -\infty, g(x)\in \mathbb{R}\},\\ \Omega_9 &:= \{x: f(x) = -\infty, g(x)\in =-\infty\},\\ \end{align} Then we note that $$ \{x : f(x) < g(x)\} = \bigcup_{j=1}^9 \Big(\Omega_j \cap \{x : f(x) < g(x)\} \Big) $$ and then it is enough to show that $\Omega_j \cap \{x : f(x) < g(x)\}$ is measurable for each $j \in \{1,...,9\}$.

Abm
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