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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?


proof


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

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

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

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

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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.

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It is certainly not a corollary, but the two results are strongly connected. Together they show that a function is measurable if and only if it is the limit of simple functions. So together, they give an alternative characterization to the definition of "measurable function". Since the foundation of the theory of integration is the simple function, this shows that the concept of "measurable function" is the right one to be looking at.

Only René could say why he called it a "corollary", but one reasonable fix would be to change it to "Theorem 8.9" and then add:

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.

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