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The PMEA three-quarter separation estimate

Statement

Assume PMEA (PMEA and PMEA-sigma). Let $X$ be a normal space (Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly), let ${F_{i} : i \in I}$ be a discrete family of subsets of $X$ (Discrete families and $σ$ -locally-finite and $σ$ -discrete bases), and for each $x \in X$ let $U_{x}$ be a downwards-directed family of neighbourhoods of $x$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open) such that $∣ U_{x} ∣ < c$ and: whenever $G \subseteq X$ is open, $i \in I$ and $x \in F_{i} \subseteq G$ , there is $U \in U_{x}$ with $U \subseteq G$ . Then there is a function $x \mapsto U^{x}$ with $U^{x} \in U_{x}$ and $U^{x} \cap U^{y} = \emptyset$ whenever $x \in F_{i}$ , $y \in F_{j}$ with $i \neq j$ .

If $X$ is first countable (First countable space: a countable neighbourhood base at every point) and only PMEA- $σ$ is assumed, the same conclusion holds for countable families $U_{x}$ of open neighbourhoods of $x$ that are local bases.

Facts & Assumptions

Given: A normal space $X$ , a discrete family ${F_{i} : i \in I}$ , families $U_{x}$ of neighbourhoods of the points as in the statement, and a full extension $ν$ of the fair-coin product measure on $2^{I}$ (PMEA and PMEA-sigma).

[F1]

Under PMEA, $ν$ may be taken $c$ -additive; under PMEA- $σ$ it may be taken countably additive. Consequently, if ${A_{t}}$ is an upwards-directed family of subsets of $2^{I}$ of cardinality $< c$ in the first case and countable in the second, covering $2^{I}$ , then $\sup_{t} ν (A_{t}) = 1$ (Fremlin, Lemma 8E and the continuity-from-below consequence of PMEA and PMEA-sigma).

[F2]

$ν$ agrees with $μ_{I}$ on cylinders; in particular, for distinct $i, j \in I$ the event ${z : z (i) \neq z (j)}$ has measure $1 / 2$ (PMEA and PMEA-sigma).

[L1]

For each $z \in 2^{I}$ , the closures of $⋃_{z (i) = 1} F_{i}$ and $⋃_{z (i) = 0} F_{i}$ are disjoint. Indeed, if a point lay in both closures, a neighbourhood meeting at most one member of the discrete family would have to meet one member from each complementary subfamily, a contradiction. Normality therefore supplies disjoint open sets containing the two original subunions (Discrete families and $σ$ -locally-finite and $σ$ -discrete bases, Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly). No individual $F_{i}$ is asserted closed.

[L2]

Probabilities are monotone, and $ν (2^{I} ∖ A) = 1 - ν (A)$ (PMEA and PMEA-sigma).

Proof

technique · direct

1.1

For $z \in 2^{I}$ choose disjoint open $G_{z}, H_{z}$ containing respectively the closures of $⋃_{z (i) = 1} F_{i}$ and $⋃_{z (i) = 0} F_{i}$ , by normality via [L1]. In particular $F_{i} \subseteq G_{z}$ when $z (i) = 1$ and $F_{i} \subseteq H_{z}$ when $z (i) = 0$ .

givenL1

2.1

For $x \in F_{i}$ and $U \in U_{x}$ put $A (x, U) : = {z : z (i) = 1 and U \subseteq G_{z}, or z (i) = 0 and U \subseteq H_{z}}$ . If $V \subseteq U$ , then $A (x, V) \supseteq A (x, U)$ . Because $U_{x}$ is downwards directed, the events $A (x, U)$ are therefore upwards directed. They cover $2^{I}$ : for $z$ with $z (i) = 1$ we have $F_{i} \subseteq G_{z}$ , so the hypothesis on $U_{x}$ gives $U \subseteq G_{z}$ , and symmetrically for $z (i) = 0$ .

step 1.1given

3.1

For $x \in F_{i}$ there is $U^{x} \in U_{x}$ with $ν (A (x, U^{x})) > 3 / 4$ : the family ${A (x, U)}$ is upwards directed, of cardinality $< c$ , and covers $2^{I}$ , so its measures have supremum $1$ by [F1].

step 2.1F1

4.1

If $x \in F_{i}$ , $y \in F_{j}$ with $i \neq j$ , then $ν (A (x, U^{x}) \cap A (y, U^{y}) \cap {z : z (i) \neq z (j)}) > 0$ : the first two complements have measure strictly below $1 / 4$ by step 3.1, while the complement of the difference event has measure $1 / 2$ by [F2]. Hence the complement of the displayed intersection has measure strictly below $1 / 4 + 1 / 4 + 1 / 2 = 1$ by subadditivity, so the intersection has positive measure.

step 3.1F2L2

5.1

Choose $z$ in that intersection. Since $z (i) \neq z (j)$ , either $z (i) = 1$ , $z (j) = 0$ , so that $U^{x} \subseteq G_{z}$ , $U^{y} \subseteq H_{z}$ and $U^{x} \cap U^{y} \subseteq G_{z} \cap H_{z} = \emptyset$ , or the reverse. Hence $U^{x} \cap U^{y} = \emptyset$ .

step 4.1step 1.1

5.2

In the PMEA- $σ$ , first countable case the same argument applies with countable local bases. Such a base is downwards directed: for $U, V$ in the base, $U \cap V$ is a neighbourhood of $x$ , so some base member lies inside it. Thus the upwards-directed countable cover ${A (x, U) : U \in U_{x}}$ has a member of measure $> 3 / 4$ by countable additivity, and steps 4.1 and 5.1 are unchanged.

step 3.1step 4.1F1

6.1

Steps 3.1 and 5.1 give the required assignment under PMEA, and step 5.2 gives it under PMEA- $σ$ for first countable $X$ .

step 3.1step 5.1step 5.2∎

Remarks

The numbers. Two events of measure above $3 / 4$ overlap in measure above $1 / 2$ , and the difference event has measure exactly $1 / 2$ ; a point of the triple overlap separates the two chosen neighbourhoods. The companion page computes this arithmetic as an example.
Only two coordinates are used, through the measure of ${z : z (i) \neq z (j)}$ .

Depends on

Used by

Dependency tree · two levels

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Sources

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