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:
$$\inf_{j\in\mathbb{N}} u_j(x) = -\sup_{j\in\mathbb{N}} u_j(-x), \tag{8.10}$$
$$\liminf_{j\to\infty} u_j(x) := \sup_{k\in\mathbb{N}} \Big( \inf_{j\geqslant k}u_j(x) \Big) = \lim_{k\to\infty} \Big( \inf_{j\geqslant k}u_j(x) \Big), \tag{8.11}$$
$$\limsup_{j\to\infty} u_j(x) := \inf_{k\in\mathbb{N}} \Big( \sup_{j\geqslant k}u_j(x) \Big) = \lim_{k\to\infty} \Big( \sup_{j\geqslant k}u_j(x) \Big), \tag{8.12}$$
Best Answer
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.