# Is Schilling's Corollary 8.9 really a corollary?

I am reading René Schilling's Measures, Integrals and Martingales and am confused as to why he considers Corollary 8.9 a corollary of Theorem 8.8, rather than a completely separate theorem (which it appears to be).

Theorem 8.8:

Let $$X$$ be a measurable space. Every $$\mathcal{A}/\bar{\mathcal{B}}$$-measurable numerical function $$u: X \to\bar{\mathbb{R}}$$ is the pointwise limit of simple functions: $$u(x) = \lim_{j\to\infty} f_j(x), f_j\in\mathcal{E}(\mathcal{A})$$ and $$|f_j|\leqslant|u|$$. If $$u\geqslant 0$$, all $$f_j$$ can be chosen to be positive and increasing towards $$u$$ so that $$u = \sup_{j\in\mathbb{N}} f_j$$.

Corollary 8.9:

Let $$X$$ be a measurable space. If $$u_j: X \to \bar{\mathbb{R}}, j\in\mathbb{N},$$ are measurable functions, then so are $$\sup_{j\in\mathbb{N}} u_j,\qquad \inf_{j\in\mathbb{N}} u_j,\qquad \limsup_{j\to\infty} u_j,\qquad \liminf_{j\to\infty} u_j,\qquad$$ and, whenever it exists, $$\lim_{j\to\infty} u_j$$.

From what I can tell, it seems that 8.9 doesn't follow from 8.8 at all. Schilling offers a proof of 8.9, which I'll insert below, but it doesn't reference anything relating to 8.8. Am I missing a key point here, or is calling this a "corollary" just a mistake?

Also for completeness, here are Eqs. 8.10–8.12 referenced in the proof:

$$\inf_{j\in\mathbb{N}} u_j(x) = -\sup_{j\in\mathbb{N}} u_j(-x), \tag{8.10}$$

$$\liminf_{j\to\infty} u_j(x) := \sup_{k\in\mathbb{N}} \Big( \inf_{j\geqslant k}u_j(x) \Big) = \lim_{k\to\infty} \Big( \inf_{j\geqslant k}u_j(x) \Big), \tag{8.11}$$

$$\limsup_{j\to\infty} u_j(x) := \inf_{k\in\mathbb{N}} \Big( \sup_{j\geqslant k}u_j(x) \Big) = \lim_{k\to\infty} \Big( \sup_{j\geqslant k}u_j(x) \Big), \tag{8.12}$$

Corollary 8.9 is not a corollary of Theorem 8.8, but it does seem to be a corollary of Lemma 8.1, which gives NASC for measurability of a function $$u:X\mapsto R$$. One sufficient condition for measurability of $$u$$ is that $$\{u>a\}\in {\cal A}$$ for every $$a\in R$$; this condition is applied in the proof of Corollary 8.9.
Corollary 8.10 A function $$u:X\to\bar{\mathbb{R}}$$ is $$\mathcal{A}/\bar{\mathcal{B}}$$-measurable if and only if it is the pointwise limit of simple functions.