TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Integrable simple functions are dense in $L^{1} (μ)$

Statement

For every $f \in L^{1} (μ)$ there is a sequence of integrable simple functions $(s_{n})$ such that $\int ∣ f - s_{n} ∣ d μ ⟶ 0.$

Facts & Assumptions

Given: An integrable function $f$ .

[L1]

Every nonnegative measurable function is the increasing limit of simple measurable functions (Every nonnegative measurable function is the increasing limit of simple measurable functions).

[L2]

Real and complex integrability are defined in Integrable real and complex functions, and their integrals.

[L3]

Dominated convergence gives $L^{1}$ convergence under an integrable majorant (Dominated convergence).

[L4]

The nonnegative integral is additive, monotone, and homogeneous (Additivity of the nonnegative Lebesgue integral, Monotonicity and nonnegative homogeneity of the nonnegative integral).

Proof

technique · direct

1.1L1L2L3construct

Suppose first that $f$ is real-valued. Choose simple $u_{n} ↑ f^{+}$ and $v_{n} ↑ f^{-}$ by [L1], and put $s_{n} : = u_{n} - v_{n}$ . Then $s_{n}$ is an integrable simple function and $∣ f - s_{n} ∣ = (f^{+} - u_{n}) + (f^{-} - v_{n}) ↓ 0,$ with $∣ f - s_{n} ∣ \leq ∣ f ∣$ . By [L3], $\int ∣ f - s_{n} ∣ d μ \to 0$ .

2.1step 1.1L2L4∎

For complex $f = u + i v$ , apply step 1.1 separately to $u$ and $v$ to obtain real simple functions $p_{n}, q_{n}$ with $\int ∣ u - p_{n} ∣ d μ \to 0, \int ∣ v - q_{n} ∣ d μ \to 0.$ Set $s_{n} : = p_{n} + i q_{n}$ . Then $s_{n}$ is a simple integrable function and $∣ f - s_{n} ∣ \leq ∣ u - p_{n} ∣ + ∣ v - q_{n} ∣ .$ By [L4], $\int ∣ f - s_{n} ∣ d μ \leq \int ∣ u - p_{n} ∣ d μ + \int ∣ v - q_{n} ∣ d μ ⟶ 0,$ so $(s_{n})$ converges to $f$ in $L^{1}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.26 (standard reference, not scraped)

Integrable simple functions are dense in L1(μ)L1(μ)