ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedaudited 2026-09-22

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Uniform null capture for a constant block function

Example

For the constant function $f (n) = 0$ , the uniform capture lemma assigns a null $G_{δ}$ set $N_{f}$ . Whenever an open $U$ of measure below one contains $N_{f}$ , the associated finite capture sets satisfy $0 \in φ_{U} (n)$ for all sufficiently large $n$ .

Verification

Given: The constant function $f : ω \to ω$ , $f (n) = 0$ , and an open set $U \supseteq N_{f}$ of coin measure below one.

[F1] Uniform null G-delta sets capture block functions: for every $f \in ω^{ω}$ there is a uniformly assigned null $G_{δ}$ set $N_{f}$ , and if an open $U$ of measure below one contains $N_{f}$ , then the finite capture sets satisfy $f (n) \in φ_{U} (n)$ for all sufficiently large $n$ .

1.1

Apply [F1] to the constant function $f (n) = 0$ . It supplies the uniformly assigned set $N_{f}$ and says directly that $N_{f}$ is a null $G_{δ}$ .

1.2

Since the given $U$ is open, has measure below one, and contains $N_{f}$ , the capture clause of [F1] gives $f (n) \in φ_{U} (n)$ for all sufficiently large $n$ . Because $f (n) = 0$ for every $n$ , this is exactly $0 \in φ_{U} (n)$ eventually.

2.1

Thus [step 1.1] gives the claimed null $G_{δ}$ , and [step 1.2] gives the claimed eventual, rather than pointwise, capture of the constant function.

step 1.1step 1.2∎

Depends on

Uniform null G-delta sets capture block functions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Hiromi Ishii, Regularity Properties and Inaccessible Cardinals (standard reference, not scraped)