Almost Everywhere Behavior of Functions According to Partition Measures
Authors: Chan William , Jackson Stephen , Trang Nam 
Published: January 2024

Abstract
This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.
The following summarizes the main results proved under suitable partition hypotheses.

•
If

$\kappa $

is a cardinal,

$\epsilon < \kappa $

,

${\mathrm {cof}}(\epsilon ) = \omega $

,

$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$

and

$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$

, then

$\Phi $

satisfies the almost everywhere short length continuity property: There is a club

$C \subseteq \kappa $

and a

$\delta < \epsilon $

so that for all

$f,g \in [C]^\epsilon _*$

, if

$f \upharpoonright \delta = g \upharpoonright \delta $

and

$\sup (f) = \sup (g)$

, then

$\Phi (f) = \Phi (g)$

.

•
If

$\kappa $

is a cardinal,

$\epsilon $

is countable,

$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$

holds and

$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$

, then

$\Phi $

satisfies the strong almost everywhere short length continuity property: There is a club

$C \subseteq \kappa $

and finitely many ordinals

$\delta _0, ..., \delta _k \leq \epsilon $

so that for all

$f,g \in [C]^\epsilon _*$

, if for all

$0 \leq i \leq k$

,

$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$

, then

$\Phi (f) = \Phi (g)$

.

•
If

$\kappa $

satisfies

$\kappa \rightarrow _* (\kappa )^\kappa _2$

,

$\epsilon \leq \kappa $

and

$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$

, then

$\Phi $

satisfies the almost everywhere monotonicity property: There is a club

$C \subseteq \kappa $

so that for all

$f,g \in [C]^\epsilon _*$

, if for all

$\alpha < \epsilon $

,

$f(\alpha ) \leq g(\alpha )$

, then

$\Phi (f) \leq \Phi (g)$

.

•
Suppose dependent choice (

$\mathsf {DC}$

),

${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$

and the almost everywhere short length club uniformization principle for

${\omega _1}$

hold. Then every function

$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$

satisfies a finite continuity property with respect to closure points: Let

$\mathfrak {C}_f$

be the club of

$\alpha < {\omega _1}$

so that

$\sup (f \upharpoonright \alpha ) = \alpha $

. There is a club

$C \subseteq {\omega _1}$

and finitely many functions

$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$

so that for all

$f \in [C]^{\omega _1}_*$

, for all

$g \in [C]^{\omega _1}_*$

, if

$\mathfrak {C}_g = \mathfrak {C}_f$

and for all

$i < n$

,

$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$

, then

$\Phi (g) = \Phi (f)$

.

•
Suppose

$\kappa $

satisfies

$\kappa \rightarrow _* (\kappa )^\epsilon _2$

for all

$\epsilon < \kappa $

. For all

$\chi < \kappa $

,

$[\kappa ]^{<\kappa }$

does not inject into

${}^\chi \mathrm {ON}$

, the class of

$\chi $

-length sequences of ordinals, and therefore,

$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$

. As a consequence, under the axiom of determinacy

$(\mathsf {AD})$

, these two cardinality results hold when

$\kappa $

is one of the following weak or strong partition cardinals of determinacy:

${\omega _1}$

,

$\omega _2$

,

$\boldsymbol {\delta }_n^1$

(for all

$1 \leq n < \omega $

) and

$\boldsymbol {\delta }^2_1$

(assuming in addition

$\mathsf {DC}_{\mathbb {R}}$

).