TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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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

1.1

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], $\int ∣ f - s_{n} ∣ d μ \to 0$ .

L1L2L3construct

2.1

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 $L^{1}$ .

step 1.1L2L4∎

Depends on

Used by

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Dependency tree · two levels

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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)