# Proof of Corollary 8.12 in Schilling's “Measures, Integrals and Martingales”?

In conjuction with this previous post of mine, I'm starting to think Rene Schilling dropped the ball in chapter 8 of his book on measure theory.

If $$u, v$$ are $$\mathcal{A}/\bar{\mathcal{B}}$$ measurable functions, then $$\{u

This is presumably a corollary of the preceding results in the chapter (Schilling offers no proof). I would almost agree that this is a corollary, because if the function $$u-v$$ is well defined, then it is also $$\mathcal{A}/\bar{\mathcal{B}}$$ measurable (by Corollary 8.10), and thus the sets above are equivalent to the sets

$$\{u-v<0\}, \;\; \{u-v\leqslant 0\}, \;\; \{u -v = 0\}, \;\; \{u -v \neq 0\},$$

which are $$\mathcal{A}/\bar{\mathcal{B}}$$ measurable by Lemma 8.1 (actually, by an extension of Lemma 8.1 to extended real–valued, $$\mathcal{A}/\mathcal{\bar{B}}$$-measureable functions).

But the if in the preceding paragraph seems important. If $$u$$ and $$v$$ both take values of $$+\infty$$ or $$-\infty$$ for the same $$x$$, then $$u-v$$ is not well defined, and the above reasoning doesn't work. Yet for these $$x$$, the expressions of the form $$\{u are still well defined, so we can't just exclude these cases.

How can we resolve this apparently incomplete gap in the above would-be proof? Or is there some other way to prove the corollary that gets around this issue entirely?

I assume that $$\bar{\mathcal{B}}=\mathcal{B}(\bar{\mathbb{R}})$$. Let $$(\Omega,\mathcal{F})$$ be a measurable space. Let $$u,v:\Omega\rightarrow\mathbb{\bar{\mathbb{R}}}$$ be $$\mathcal{F}/\bar{\mathcal{B}}$$-measurable functions. Let $$A=\{\omega\in\Omega\mid u(\omega). We go to show that $$A\in\mathcal{F}$$. Observe that $$A=\cup_{r\in\mathbb{Q}}\{\omega\mid u(\omega) For each $$r\in\mathbb{Q}$$, $$\{\omega\mid u(\omega), $$\{\omega\mid r are elements in $$\mathcal{F}$$. It follows that $$A\in\mathcal{F}$$ because the union $$\cup_{r\in\mathbb{Q}}$$ is a countable union.
Similarly, $$[u>v]\in\mathcal{F}$$. Finally, $$[u=v] = \Omega\setminus([uv])\in\mathcal{F}$$.