ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-09-01

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A concrete countable dense family in $L^{2} [0, 1]$

Example

Assume the Axiom of Countable Choice.

Inside $L^{2} ([0, 1])$ , the finite rational linear combinations of indicators of rational half-open intervals $(a, b] \subseteq [0, 1]$ form a countable dense family.

Facts & Assumptions

Given: The Axiom of Countable Choice and the space $L^{2} ([0, 1])$ .

[L1]

Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for every finite $p$ (Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for $1 \leq p < \infty$ ).

Verification

technique · direct

1.1

Specialize [L1] to $n = 1$ and $p = 2$ . The rational boxes in one dimension are [L1, given] the rational half-open intervals $(a, b]$ , and restricting to those contained in $[0, 1]$ still leaves a countable family.

L1given

2.1

Extend a function on $[0, 1]$ by $0$ outside $[0, 1]$ . Then approximation in [step 1.1, algebra] $L^{2} (R)$ by rational interval step functions supported in $[0, 1]$ restricts back to approximation in $L^{2} ([0, 1])$ .

step 1.1algebra

3.1

Therefore the stated family is an explicit countable dense subset of [step 2.1] $L^{2} ([0, 1])$ .

step 2.1∎

Depends on

Rational box-step functions form a countable dense subset of $L^p(\mathbb{R}^n)$ for $1 \le p < \infty$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis (standard reference, not scraped)

A concrete countable dense family in L2[0,1]L2[0,1]