ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedaudited 2026-09-22

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The three-quarter event calculation in the PMEA proof

Example

Let $ν$ be a full extension of a fair-coin product measure on $2^{I}$ (PMEA and PMEA-sigma) and let $E, F, D \subseteq 2^{I}$ be sets with $ν (E) > 3 / 4$ , $ν (F) > 3 / 4$ and $ν (D) = 1 / 2$ , where $D = {z : z (i) \neq z (j)}$ is the difference event of two distinct coordinates $i \neq j$ . The example computes that $ν (E \cap F) > 1 / 2$ and $ν (E \cap F \cap D) > 0$ . This is only the measure-theoretic calculation consumed by the separation lemma; the lemma's additional definitions relate its good events to disjoint neighbourhoods.

Facts & Assumptions

Given: A probability $ν$ on the full power set of $2^{I}$ , sets $E, F$ with $ν (E), ν (F) > 3 / 4$ , and a set $D$ with $ν (D) = 1 / 2$ .

[F1]

Probability and complement: $ν (A^{c}) = 1 - ν (A)$ for every $A$ , and $ν (2^{I}) = 1$ (PMEA and PMEA-sigma).

[L1]

Subadditivity for two or three sets: $ν (A \cup B) \leq ν (A) + ν (B)$ , hence for three sets as well (PMEA and PMEA-sigma).

[L2]

For distinct coordinates $i \neq j$ the difference event has $ν ({z : z (i) \neq z (j)}) = 1 / 2$ (PMEA and PMEA-sigma).

Verification

technique · direct

1.1

$ν (E \cap F) > 1 / 2$ : by [F1] and [L1], $1 = ν (E \cup F) + ν ((E \cup F)^{c}) \leq ν (E) + ν (F) + ν (E^{c} \cap F^{c})$ ; hence $ν (E \cap F) = 1 - ν (E^{c} \cup F^{c}) \geq 1 - (ν (E^{c}) + ν (F^{c})) > 1 - (1 / 4 + 1 / 4) = 1 / 2$ , since $ν (E^{c}) < 1 / 4$ and $ν (F^{c}) < 1 / 4$ by [F1].

givenF1L1

2.1

$ν (E \cap F \cap D) > 0$ : the complement of the triple intersection is contained in $E^{c} \cup F^{c} \cup D^{c}$ , which has measure at most $ν (E^{c}) + ν (F^{c}) + ν (D^{c}) < 1 / 4 + 1 / 4 + 1 / 2 = 1$ by [L1] and [F1] (using $ν (D^{c}) = 1 / 2$ ). Hence the triple intersection has positive measure and is nonempty.

step 1.1F1L1L2

3.1

Hence there exists $z \in E \cap F \cap D$ ; every such $z$ lies in both $E$ and $F$ and satisfies $z (i) \neq z (j)$ by the definition of $D$ . This example proves no topological conclusion from the abstract sets $E$ and $F$ : in The PMEA three-quarter separation estimate the separately defined good events and separating open sets give that conclusion.

step 2.1L2∎

Remarks

Strictness matters. Both good events have measure strictly above $3 / 4$ , so the complement of the triple intersection has measure strictly below $1$ ; with $\geq 3 / 4$ the conclusion $ν (E \cap F \cap D) > 0$ could fail.
Only two coordinates are used, through $ν (D) = 1 / 2$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

D. H. Fremlin, Real-valued-measurable cardinals (standard reference, not scraped)