CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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Every measurable function admits simple approximations dominated by its absolute value

Statement

Let $(X, A)$ be a measurable space and let $f : X \to \overline{R}$ be measurable. Then there is a sequence of simple functions $s_{k} : X \to R$ such that

$∣ s_{k} ∣ \leq ∣ f ∣ for every k,$

and $s_{k} (x) \to f (x)$ for every $x \in X$ .

Facts & Assumptions

Given: A measurable function $f : X \to \overline{R}$ .

[L1]

The positive and negative parts satisfy $f = f^{+} - f^{-}$ , $∣ f ∣ = f^{+} + f^{-}$ , and at each point at least one of $f^{+}, f^{-}$ is zero. (The positive and negative parts of a function)

[L2]

The arithmetic-and-lattice theorem makes $f^{+}$ and $f^{-}$ measurable. (Arithmetic and lattice operations preserve measurability whenever they are defined)

[L3]

Every nonnegative measurable function admits an increasing sequence of simple approximations. (Every nonnegative measurable function admits an explicit increasing sequence of simple approximations)

Proof

technique · direct

1.1

By [L2], the functions $f^{+}$ and $f^{-}$ are measurable and nonnegative. Applying [L1, L2, L3] [L3] to them gives simple functions $u_{k} ↑ f^{+}$ and $v_{k} ↑ f^{-}$ .

L1L2L3

2.1

Put $s_{k} : = u_{k} - v_{k}$ . Because $u_{k}$ and $v_{k}$ are simple, $s_{k}$ is a [step 1.1, L1] simple real-valued function. At each point, [L1] makes at least one of $u_{k}$ and $v_{k}$ equal to $0$ , so

$∣ s_{k} ∣ = u_{k} + v_{k} \leq f^{+} + f^{-} = ∣ f ∣ .$

Also $s_{k} \to f^{+} - f^{-} = f$ pointwise because $u_{k} \to f^{+}$ and $v_{k} \to f^{-}$ . [step 1.1, L1] ∎

Depends on

Used by

A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Measure, Integration and Real Analysis, Theorem 2.89 (standard reference, not scraped)