On sequences of homomorphisms into measure algebras and the Efimov Problem
aa r X i v : . [ m a t h . L O ] J a n ON SEQUENCES OF HOMOMORPHISMS INTO MEASUREALGEBRAS AND THE EFIMOV PROBLEM
PIOTR BORODULIN–NADZIEJA AND DAMIAN SOBOTA
Abstract.
For given Boolean algebras A and B we endow the space H ( A , B ) of all Boolean homomorphisms from A to B with various topologies and studyconvergence properties of sequences in H ( A , B ) . We are in particular interested inthe situation when B is a measure algebra as in this case we obtain a natural toolfor studying topological convergence properties of sequences of ultrafilters on A in random extensions of the set-theoretical universe. This appears to have strongconnections with Dow and Fremlin’s result stating that there are Efimov spaces inthe random model. We also investigate relations between topologies on H ( A , B ) fora Boolean algebra B carrying a strictly positive measure and convergence propertiesof sequences of measures on A . Introduction
It is a common issue in analysis and topology to compare convergence propertiesof sequences in topological spaces, in particular, to study when a sequence conver-gent with respect to one topology converges also with respect to another one. Forinstance, one can ask what properties a Banach space must have so that its ev-ery weakly convergent sequence is also norm convergent (studying so called Schurproperty), or investigate when a given topological space does not contain non-trivialconvergent sequences, i.e. when every convergent sequence is eventually constant.In this paper we also deal with this kind of problems—our setting are spaces of ho-momorphisms between Boolean algebras, especially measure algebras, endowed withseveral miscellaneous topologies.There are several motivations standing behind our research, but before we presentthem, together with our main results, we introduce some basic notations. For acardinal κ let M κ denote the measure algebra of the standard product measure λ κ onthe product space κ . If κ = 1 , then put = M . If A is a Boolean algebra, then by H ( A , M κ ) we denote the family of all Boolean homomorphisms from A into M κ . Themeasure λ κ induces the so-called Fréchet–Nikodym metric d λ κ on M κ , and thus M κ may be treated as a metric topological space. Since H ( A , M κ ) is simply a subset ofthe family of all functions from A into M κ , this further leads to endowing H ( A , M κ ) with two natural topologies— the pointwise topology and the uniform topology . In thisarticle we will investigate these structures and their variations as well as the relationsbetween them. Mathematics Subject Classification.
Primary: 28A20, 28A60, 03E40. Secondary: 28E15,03E75, 28A33.
Key words and phrases.
Efimov problem, measure algebras, weak convergence of measures, alge-braic convergence, random forcing, perfect Hamming codes.The first author was supported by National Science Center project no. 2018/29/B/ST1/00223.The second author was supported by the Austrian Science Fund, grant M2500-N35.
Our first motivation for this research is the observation that for κ > the space H ( A , M κ ) with the pointwise topology may be treated as a natural generalizationof the Stone space of the Boolean algebra A , or a Stone space of A of higher order.Indeed, recall that every ultrafilter on the Boolean algebra A induces a homomorphismfrom A into the two-element algebra , and vice versa , and thus the Stone space St ( A ) of A may be seen as the space H ( A , ) , endowed with the pointwise topologyoriginating from the Fréchet–Nikodym metric d λ , induced by the trivial measure λ on . As the literature concerning convergence properties of sequences in Stone spacesof Boolean algebras, or more generally in totally disconnected topological spaces, isquite vast, cf. e.g. [BD19], [Har07], [HS20], we were motivated to conduct similarresearch in the setting of the spaces H ( A , M κ ) for general κ ’s and various topologies.Note that similar generalizations of topological spaces are quite common in analysis,e.g. one can think of Radon measures of norm on compact spaces as generalizationsof points in those spaces, or, as it is done in non-commutative topology, of projectionsin general C*-algebras as being analogous to characteristic functions of clopen subsetsof locally compact spaces.For every κ the uniform topology on the space H ( A , M κ ) is of course finer than thepointwise topology, hence every uniformly convergent sequence in H ( A , M κ ) is point-wise metric convergent. As one can suspect, the converse in general does not hold—cf.Section 6 for relevant counterexamples (especially interesting is the example describedin Section 6.4, to obtain which we used tools from information theory such as perfectHamming codes). In Section 4.3 we introduce and study a strengthening of pointwisemetric convergence—called by us the pointwise algebraic convergence —that impliesunder certain conditions put on the Boolean algebra A (such as σ -completeness) alsothe uniform convergence (Corollary 8.4). This mode of convergence is related tothe well-known notion of the algebraic convergence in σ -complete Boolean algebras,studied e.g. in [BGJ98], [BFH99], [Jec18].In Section 4.4, using the dual functions to homomorphisms from A into M κ andBorel subsets of the Stone space A , we endow the space H ( A , M κ ) with yet anothertopology, which we call the pointwise Borel metric topology . This topology is finer thanthe topology of pointwise convergence and has turned out to be useful in investigationsrelated to our second motivation which is the fact that each homomorphism ϕ ∈H ( A , M κ ) naturally induces a measure on A of the form λ κ ◦ ϕ , as well as, by thevirtue of the celebrated Maharam theorem, for each strictly positive measure µ on A there are a cardinal γ and a (usually not unique) homomorphism ψ ∈ H ( A , M γ ) such that µ = ψ ◦ λ γ . This correspondence has inspired us to study relations betweensequences ( ϕ n ) of homomorphisms in H ( A , M κ ) and sequences of measures of theform λ κ ◦ ϕ n . In particular, we show in Corollary 5.2 connections between the uniformconvergence, pointwise Borel metric convergence, and pointwise metric convergence ofa sequence ( ϕ n ) in H ( A , M κ ) and the norm convergence, weak convergence, and weak*convergence of the sequence ( λ κ ◦ ϕ n ) of measures on A , respectively. In Section 5we prove also that whenever A has the Grothendieck property (see the end of Section3.1 for the definition), then every pointwise metric convergent sequence in H ( A , M κ ) is pointwise Borel metric convergent—a fact that does not hold for general Booleanalgebras A (cf. the example in Section 6.1).Our last main motivation lies in the theory of forcing, where actually our interestin studying the subject of this article started. Imagine that we force with a Booleanalgebra B over the ground model V and that we are interested in the behaviour of N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 3 sequences of ultrafilters in the forcing extension V B which are defined on a fixedground model Boolean algebra A . This is actually a quite common situation—e.g.reals in the extension can be seen as ultrafilters on the Cantor algebra C , i.e. thefree countable Boolean algebra. The usual problem is that the access to objects inforcing extensions is quite remote , through B -names only. However, the B -namesfor ultrafilters on A naturally correspond to homomorphisms ϕ : A → B (see Section7). Then, the convergence of ultrafilters (treated as elements of the Stone spaceof A in V B ) appears to be strongly connected to certain convergence properties ofsequences of homomorphisms in the ground model. Namely, we prove in Proposition7.3 that for a given sequence ( ˙ U n ) of M κ -names for ultrafilters on A the sequence ofthe corresponding homomorphisms ( ϕ ˙ U n ) in H ( A , M κ ) is algebraically convergent ifand only if the sequence (( ˙ U n ) G ) of the interpretations converges in V [ G ] for every M κ -generic filter G . It turns out that this proposition has also variants regardinguniform convergence of sequences of homomorphisms—Proposition 7.4 asserts thatif (( ˙ U n ) G ) is eventually constant in every generic extension V [ G ] , then the sequence ( ϕ ˙ U n ) is uniformly convergent. The converse holds partially—in Theorem 7.5 we showthat if a sequence of homomorphisms ( ϕ n ) in H ( A , M κ ) is uniformly convergent tosome ϕ ∈ H ( A , M κ ) , then for almost all n ∈ ω and the corresponding names ˙ U ϕ n and ˙ U ϕ for ultrafilters on A there is a condition p n ∈ M κ forcing that ˙ U ϕ n = ˙ U ϕ .This latter theorem is proved with the aid of Theorem 7.6 asserting that if for two M κ -names ˙ U and ˙ V for ultrafilters on A it holds (cid:13) M κ ˙ U 6 = ˙ V , then there exists a large condition p ∈ M κ , where large means of measure as close to / as one wishes , andan element A ∈ A such that p (cid:13) A ∈ ˙ U △ ˙ V —a result interesting on its own, since atypical argument based on the countable chain condition of M κ provides us only witha countable antichain { p n : n ∈ ω } of conditions witnessing that the two ultrafiltersare different, but with no control over the value of λ κ ( p n ) for any n ∈ ω . The proofof Theorem 7.6 is also interesting, because it boils down to purely combinatorialProposition 7.7 having a nice real-life interpretation, see Remark 7.12.The aforementioned results have connection with the famous long-standing openquestion, called the Efimov problem , asking whether there exists an Efimov space ,that is, an infinite compact Hausdorff space which does not contain copies of βω ,the Čech–Stone compactification of ω , nor non-trivial convergent sequences. No ZFCexample of an Efimov space is known, however several consistent examples have beenobtained either under some additional set-theoretic assumptions, e.g. by Fedorchuk[Fed76] (under Jensen’s diamond principle), Dow and Pichardo-Mendoza [DPM09](under the Continuum Hypothesis), Dow and Shelah [DS13] (under Martin’s axiom),or using forcing, see e.g. Sobota and Zdomskyy [SZ19] or Dow and Fremlin [DF07].Results in the latter paper are of particular interest to us as they concern forcing with M κ ; namely, Dow and Fremlin proved that if A is a ground model Boolean algebrasuch that its Stone space is an F-space, which occurs e.g. in the case of σ -completeBoolean algebras or the algebra P ( ω ) /F in , then, in V M κ , the Stone space St ( A ) does not have any non-trivial convergent sequences (which yields, e.g., that if V isa model of set theory satisfying the Continuum Hypothesis, then St ( P ( ω ) ∩ V ) isan Efimov space in any M ω -generic extension of V ). In Section 8 we prove that M κ forces that the Stone space of a given ground model Boolean algebra A doesnot contain any non-trivial convergent sequences if and only if in V every pointwisealgebraically convergent sequence in H ( A , M κ ) is uniformly convergent (Theorem 8.2). PIOTR BORODULIN–NADZIEJA AND DAMIAN SOBOTA
This, together with Dow and Fremlin’s theorem, implies that if the Stone space of aninfinite Boolean algebra A is an F-space, then every pointwise algebraically convergentsequence in H ( A , M κ ) is uniformly convergent—note that this result does not holdfor ’simple’ Boolean algebras such as the Cantor algebra, see the example in Section6.5.The structure of the paper is as follows. In the next section we present basicnotations, terminology and facts used in the paper. In Section 4 we introduce fourtypes of topologies and modes of convergence of sequences of homomorphisms, mostlyinto measure algebras. Section 5 is devoted to study relations between convergenceproperties of sequences ( ϕ n ) of homomorphisms into a Boolean algebra carrying astrictly positive measure µ and convergence properties of sequences of measures ofthe form ( µ ◦ ϕ n ) . In Section 6 we present a series of examples of sequences of homo-morphisms between Boolean algebras being convergent with respect to one topologybut not with respect to another one. In Section 7 we study relations between vari-ous types of convergence of sequences of homomorphisms into measure algebras andconvergence of corresponding ultrafilters in random generic extensions of the groundmodel. This study is continued in Section 8, where we characterize those Booleanalgebras whose Stone spaces contain no non-trivial convergent sequences in randomextensions with the aid of convergence properties of sequences of homomorphisms inthe ground model. The last section provides several open questions and problems.2. Acknowledgements
We would like to thank Grzegorz Plebanek and Krzysztof Majcher for valuablediscussions concerning homomorphisms, Hamming codes and painting fences. Wealso thank Lyubomyr Zdomskyy for reading the first draft of the paper and sharingwith us many relevant comments.3.
Notations and terminology
All compact spaces considered in the paper are assumed to be Hausdorff. If K isa compact space, then by Clopen ( K ) and Bor ( K ) we denote the Boolean algebraof clopen subsets of K and the σ -field of all Borel subsets of K , respectively. If A is a Boolean algebra, then by St ( A ) we denote its Stone space. Note that A and Clopen ( St ( A )) are isomorphic.Unless otherwise stated, all measures considered by us are probability measures. Ameasure on a Boolean algebra is always meant to be finitely additive. On the otherhand, a measure on a compact space is always a Radon measure, i.e. it is countablyadditive, Borel and inner regular with respect to compact subsets. Note that everymeasure µ on a Boolean algebra A has a unique extension to a Radon measure b µ on the Stone space St ( A ) , i.e. µ = b µ ↾ Clopen ( St ( A )) , where A is identified with Clopen ( St ( A )) . When there should be no confusion, we will usully drop b and writesimply µ for b µ .If A and B are two Boolean algebras, then H ( A , B ) denotes the family of all homo-morphisms from A to B . In Section 4 we will endow H ( A , B ) with various topologiesand consider various types of convergences of sequences in H ( A , B ) . If ϕ ∈ H ( A , B ) ,then the function f ϕ : St ( B ) → St ( A ) defined as f ϕ ( x ) = ϕ − [ x ] is continuous (here x is an ultrafilter on B ). On the other hand, if f : St ( B ) → St ( A ) is a continuousfunction, then the function ϕ f : A → B defined by the formula ϕ f ( A ) = f − [ A ] is N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 5 a homomorphism between Boolean algebras A and B . It follows that f = f ( ϕ f ) and ϕ = ϕ ( f ϕ ) .3.1. Topologies on spaces of measures.
Let A be a Boolean algebra and let K bea compact space. The space of all measures on K is denoted by P ( K ) . By C ( K ) wedenote the Banach space of continuous real-valued functions on K endowed with thesupremum topology. If µ ∈ P ( K ) and f ∈ C ( K ) , then µ ( f ) is defined as the integral µ ( f ) = R K f dµ .We endow P ( K ) with three topologies: the norm topology, the weak topology andthe weak* topology. The norm topology on P ( K ) is induced by the variation metric d var on P ( K ) defined by the formula d var ( µ, ν ) = sup A,B ∈ Bor ( K ) A ∩ B = ∅ (cid:0) | µ ( A ) − ν ( A ) | + | µ ( B ) − ν ( B ) | (cid:1) for every µ, ν ∈ P ( K ) . Note that if K is totally-disconnected, then in the aboveformula for d var ( µ, ν ) we may confine ourselves only to pairs of disjoint clopen subsetsof K . The weak topology on the dual space C ( K ) ∗ (and hence on P ( K ) ) is the weakesttopology which makes all functionals from C ( K ) ∗ continuous and so it is induced bythe subbase given by sets of the form V ( µ, ϕ, ε ) = (cid:8) ν ∈ P ( K ) : | ϕ ( µ ) − ϕ ( ν ) | < ε (cid:9) , where µ ∈ P ( K ) , ϕ ∈ C ( K ) ∗∗ and ε > . Similarly, the weak* topology on P ( K ) isdefined by sets of the form V ( µ, f, ε ) = (cid:8) ν ∈ P ( K ) : | µ ( f ) − ν ( f ) | < ε (cid:9) , where µ ∈ P ( K ) , f ∈ C ( K ) and ε > . Recall that P ( K ) with the weak* topologyis a compact space. Of course, the weak* topology on P ( K ) is weaker than the weaktopology which is on the other hand weaker than the norm topology.The following facts are well-known. Proposition 3.1.
Let K be a totally disconnected compact space. Let V be the col-lection of the sets of the form: V ( µ, C, ε ) = (cid:8) ν ∈ P ( K ) : | ν ( C ) − µ ( C ) | < ε (cid:9) , where µ ∈ P ( K ) , C is a clopen subset of K and ε > . Then V is a subbase of theweak* topology on P ( K ) . Proposition 3.2. [Die84, Theorem 11 in Section VII]
Let K be a compact space.Then the sequence ( µ n ) of measures on K converges weakly to some µ if and only if µ n ( B ) converges to µ ( B ) for each Borel subset B ⊆ K . We say that a Boolean algebra A has the Grothendieck property if every weak*convergent sequence of (signed) Radon measures on St ( A ) is weakly convergent. Itis well-known that σ -complete Boolean algebras have the property, but countableones do not (or more generally those algebras whose Stone spaces contain non-trivialconvergent sequences). PIOTR BORODULIN–NADZIEJA AND DAMIAN SOBOTA
Measure algebras. If µ is a measure on a Boolean algebra A , then we put N µ = (cid:8) A ∈ A : µ ( A ) = 0 (cid:9) . Similarly, if µ is a Radon measure on a compact space K , then we denote N µ = (cid:8) A ∈ Bor ( K ) : µ ( A ) = 0 (cid:9) . For an infinite cardinal number κ , by M κ we will denote the (standard) measure algebra of Maharam type κ , i.e. M κ = Bor(2 κ ) / N κ , where N κ is the σ -ideal of null sets with respect to the standardproduct measure λ κ on the space κ , i.e. N κ = N λ κ . By λ and M we mean simply λ ω and M ω , respectively.By we mean the 2-point Boolean algebra { , } . Notice that = M , i.e. it is ameasure algebra (with measure λ ). C will denote the Cantor algebra , i.e. the free algebra generated by ω generators.Note that C is isomorphic to the algebra of clopen subsets of the Cantor space ω andhence St ( C ) is homeomorphic to ω . We will use this identification frequently.In general we denote elements of Boolean algebras, including M κ ’s, with capitalletters. However, speaking about an element of M κ sometimes we understand it as acondition of the forcing notion M κ \ { } . In this case, we will rather use the standardforcing notation: p , q and so on.3.3. Metric Boolean algebras.
We will say, after Kolmogorov [Kol95], that aBoolean algebra B is metric if it supports a strictly positive measure, i.e. thereis a measure µ on B such that µ ( A ) > for every A ∈ B \ { } . Note that a metricalgebra need not to be σ -complete, so it is not necessarily a measure algebra. Theword metric is explained by the following simple fact. Fact 3.3. If µ is a strictly positive measure on a Boolean algebra B , then the function d µ : B × B → R defined for every A, B ∈ B by the formula d µ ( A, B ) = µ ( A △ B ) is a metric on B (called Fréchet–Nikodym metric ). Note that we may define easily a Radon version of the Fréchet–Nikodym metric.Namely, if µ is a strictly positive measure on a Boolean algebra B , then the function d Borµ : Bor ( St ( B )) × Bor ( St ( B )) → R defined as d Borµ ( A, B ) = b µ ( A △ B ) , where A, B ∈ Bor ( St ( B )) , is a pseudometric. Call d Borµ the Borel Fréchet–Nikodympseudometric on ( B , µ ) .Recall that there is a natural and well-studied notion of convergence in completeBoolean algebras. Let A be a complete Boolean algebra and let ( A n ) be a sequenceof its elements. We say that ( A n ) algebraically converges to A ∈ A if _ n ^ m>n A m = ^ n _ m>n A m = A. Such a notion was deeply studied e.g. in [BGJ98], [BFH99], [Jec18].
The sequentialtopology on A is the largest topology with respect to which all the algebraically con-vergent sequences converge. If A is a measure algebra, then the sequential topologycoincides with the topology introduced by the Fréchet–Nikodym metric (see [BGJ98]).4. Convergences and topologies in H ( A , B ) In the next subsections we will see that for given Boolean algebras A and B wemay endow the space H ( A , B ) with several topologies and convergences. N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 7
Pointwise metric topology.
Let ( B , µ ) be a metric algebra. Then, there isa natural topology on H ( A , B ) given by the metric d µ , which we call the pointwisemetric topology . Namely, an element of the subbase is of the form V ( ϕ, A, ε ) = (cid:8) ψ ∈ H ( A , B ) : d µ ( ϕ ( A ) , ψ ( A )) < ε (cid:9) for ϕ ∈ H ( A , B ) , A ∈ A and ε > .Let ( ϕ n ) be a sequence of homomorphisms from A to B . It follows that ( ϕ n ) isconvergent in the pointwise metric topology to some ϕ ∈ H ( A , B ) if and only if forevery A ∈ A and every ε > there is N ∈ ω such that d µ (cid:0) ϕ n ( A ) , ϕ ( A ) (cid:1) < ε for every n > N —in this case, we will say that ( ϕ n ) is pointwise metric convergent to ϕ . Notethat if ( ϕ n ) is not convergent to ϕ , then there are A ∈ A , ε > and a subsequence ( ϕ n k ) such that for every k ∈ ω we have ε ≤ d µ (cid:0) ϕ n k ( A ) , ϕ ( A ) (cid:1) = µ (cid:0) ϕ n k ( A ) \ ϕ ( A ) (cid:1) + µ (cid:0) ϕ ( A ) \ ϕ n k ( A ) (cid:1) , so either µ (cid:0) ϕ n k ( A ) \ ϕ ( A ) (cid:1) ≥ ε/ for infinitely many k ∈ ω , or µ (cid:0) ϕ ( A ) \ ϕ n k ( A ) (cid:1) ≥ ε/ for infinitely many k ∈ ω . Since µ (cid:0) ϕ ( A ) \ ϕ n k ( A ) (cid:1) = µ (cid:0) ϕ n k ( A c ) \ ϕ ( A c ) (cid:1) , it follows that if ( ϕ n ) does not converge pointwise metric to ϕ , then we may alwaysfind A ∈ A , ε > and a subsequence ( ϕ n k ) such that µ (cid:0) ϕ n k ( A ) \ ϕ ( A ) (cid:1) ≥ ε . We willuse this observation regularly. Remark 4.1.
Using the natural bijection between the set of ultrafilters on a Booleanalgebra A and the set of homomorphisms from H ( A , ) , we see that the pointwisemetric topology coincides with the Stone topology on the set of ultrafilters on A ,i.e. St ( A ) and H ( A , ) are homeomorphic. Thus, since embeds into every (metric)Boolean algebra B , the space H ( A , B ) with the pointwise metric topology alwayscontains a closed copy of the Stone space of A , which is immediately yielded by thefollowing trivial fact. Lemma 4.2.
Let A , B and C be Boolean algebras. Assume that ( B , µ ) and ( C , ν ) are metric and such that B is a subalgebra of C with µ = ν ↾ B . Endow H ( A , B ) and H ( A , C ) with the pointwise metric topologies. If for every A ∈ C \ B there is ε > such that d ν ( A, B ) < ε for no B ∈ B , then H ( A , B ) is closed in H ( A , C ) . Uniform topology.
Let us again assume that ( B , µ ) is a metric algebra. Wemay define a metric d hom on the set H ( A , B ) in the following way: d hom ( ϕ, ψ ) = sup (cid:8) d µ ( ϕ ( A ) , ψ ( A )) : A ∈ A (cid:9) , where ϕ, ψ ∈ H ( A , B ) . According to the topology induced by d hom , which we call the uniform topology , a sequence ( ϕ n ) converges to ϕ iff it converges uniformly withrespect to the metric d µ , i.e. for each ε > there is N such that for every n > N wehave d µ (cid:0) ϕ n ( A ) , ϕ ( A ) (cid:1) < ε for each A ∈ A . We will say in this case that ( ϕ n ) convergesuniformly to ϕ . Of course, if a sequence of homomorphisms converges uniformly, thenit converges pointwise—the converse however does not necessarily hold, see Sections6.1, 6.3 and 6.4 for appropriate examples. Remark 4.3.
Contrary to the pointwise metric topology (see Remark 4.1), for everyBoolean algebra A the uniform topology on H ( A , ) is discrete. PIOTR BORODULIN–NADZIEJA AND DAMIAN SOBOTA
Pointwise algebraic convergence.
This kind of convergence can be intro-duced in H ( A , B ) if B is σ -complete (but not necessarily metric). We say that asequence of homomorphisms ( ϕ n ) from A to B converges pointwise algebraically to ϕ if for every A ∈ A we have _ n ^ m>n ϕ m ( A ) = ^ n _ m>n ϕ m ( A ) = ϕ ( A ) . In fact, by the de Morgan laws, ( ϕ n ) converges pointwise algebraically to ϕ if andonly if any of the following two equivalent conditions hold: • for every A ∈ A we have W n V m>n ϕ m ( A ) = ϕ ( A ) ; • for every A ∈ A we have V n W m>n ϕ m ( A ) = ϕ ( A ) .If we assume that B is metric (with a strictly positive measure µ ), then if ( ϕ n ) converges pointwise algebraically to ϕ , then for each A ∈ A we have lim n →∞ µ (cid:0) ^ m>n ϕ m ( A ) (cid:1) = lim n →∞ µ (cid:0) _ m>n ϕ m ( A ) (cid:1) = µ ( ϕ ( A )) . The following fact binds the pointwise algebraic convergence with the pointwisemetric convergence. Recall that for every κ the measure λ κ is continuous , i.e. forevery decreasing sequence ( A n ) in M κ we have lim n →∞ λ κ ( A n ) = λ κ (cid:0) V n ∈ ω A n (cid:1) . Proposition 4.4.
Let A be a Boolean algebra and κ a cardinal number. Let ( ϕ n ) be a sequence of homomorphisms from A to M κ . If ( ϕ n ) is pointwise algebricallyconvergent to some ϕ ∈ H ( A , M κ ) , then ϕ n converges pointwise metric to ϕ .Proof. Let A ∈ A . We will show that lim n →∞ λ κ (cid:0) ϕ n ( A ) △ ϕ ( A ) (cid:1) = 0 . For thesake of contradiction assume that there is ε > and a subsequence ( ϕ n k ) such that λ κ ( ϕ n k ( A ) \ ϕ ( A )) ≥ ε for every k ∈ ω . Then, by the continuity of λ κ , λ κ (cid:16) ^ k ∈ ω _ l>k (cid:0) ϕ n l ( A ) \ ϕ ( A ) (cid:1)(cid:17) ≥ ε. However, since V k ∈ ω W l>k ϕ n l ( A ) = ϕ ( A ) , we get that V k ∈ ω W l>k (cid:0) ϕ n l ( A ) \ ϕ ( A ) (cid:1) = 0 ,so λ κ (cid:16) ^ k ∈ ω _ l>k (cid:0) ϕ n l ( A ) \ ϕ ( A ) (cid:1)(cid:17) ≥ ε > , a contradiction. (cid:3) As we will see in Sections 6.2 and 6.3, the converse to Proposition 4.4 does nothold in general. However, for B = we have the following corollary. Corollary 4.5.
Let A be a Boolean algebra. Then, in H ( A , ) the pointwise algebraicconvergence coincides with the pointwise metric convergence.Proof. Let ( ϕ n ) be a sequence in H ( A , ) . If ( ϕ n ) is pointwise algebraically conver-gent, then by Proposition 4.4 it is also pointwise metric convergence.Conversely, if ( ϕ n ) is pointwise metric convergent to ϕ ∈ H ( A , ) , then for every A ∈ A there is N ∈ ω such that ϕ n ( A ) = ϕ ( A ) for every n > N . It follows that V n>N ϕ n ( A ) = ϕ ( A ) , so in particular W N ∈ ω V n>N ϕ n ( A ) = ϕ ( A ) . Hence, ( ϕ n ) converges pointwise algebraically to ϕ . (cid:3) N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 9
Pointwise Borel metric topology and uniform Borel topology.
Let A bea Boolean algebra and ( B , µ ) a metric Boolean algebra. To introduce yet anothertopology on H ( A , B ) , called by us pointwise Borel metric topology , we need to appealto dual continuous functions and the pseudometric d Borµ . Namely, we define thesubbase of the topology to be given by the sets of the form V Bor ( ϕ, A, ε ) = (cid:8) ψ ∈ H ( A , B ) : d Borµ (cid:0) f − ϕ [ A ] , f − ψ [ A ] (cid:1) < ε (cid:9) for ϕ ∈ H ( A , B ) , A ∈ Bor ( St ( A )) and ε > .Let ( ϕ n ) be a sequence of homomorphisms from A to B . For each n ∈ ω let f n = f ϕ n and let f = f ϕ . We say that ( ϕ n ) converges pointwise Borel metric to ϕ if lim n →∞ b µ (cid:0) f − n [ B ] △ f − [ B ] (cid:1) = 0 for every B ∈ Bor ( St ( A )) . Obviously, if a sequenceof homomorphisms converges pointwise Borel metric, then it converges pointwise met-ric, however the converse may not hold—we provide relevant counterexamples in Sec-tions 6.1 and 6.3 as well as some positive results in Section 5 (see, in particular,Corollary 5.6).Similarly to the uniform topology on H ( A , B ) , we may define the uniform Boreltopology . Namely, we define a Borel version d Borhom of the metric d hom : d Borhom ( ϕ, ψ ) = sup (cid:8) d Borµ (cid:0) f − ϕ [ A ] , f − ψ [ A ] (cid:1) : A ∈ Bor ( St ( A )) (cid:9) , where ϕ, ψ ∈ H ( A , B ) . Note that, since µ is strictly positive on A , d Borhom is a metric,even though d Borµ is only a pseudometric. As before, we will say that a sequence ( ϕ n ) converges uniformly Borel to ϕ if for every ε > there is N ∈ ω such that d Borhom ( ϕ n , ϕ ) < ε for every n > N , i.e. b µ (cid:0) f − n [ B ] △ f − [ B ] (cid:1) < ε for every n > N andevery Borel subset B ⊆ St ( A ) . Of course, if a sequence converges uniformly Borel,then it converges pointwise Borel metric and uniformly. It appears that, conversely,the uniform convergence easily implies the Borel uniform convergence, because, infact, the uniform topology and uniform Borel topology coincide. Lemma 4.6.
Let A be a Boolean algebra and ( B , µ ) a metric Boolan algebra. Then,for the metrics d hom and d Borhom on H ( A , B ) we have d hom = d Borhom .Proof.
Fix ϕ, ψ ∈ H ( A , B ) . Since every clopen set in St ( A ) is Borel, we have d hom ( ϕ, ψ ) ≤ d Borhom ( ϕ, ψ ) . Let ε > —we will show that d Borhom ( ϕ, ψ ) ≤ d hom ( ϕ, ψ ) + ε , which willprove that d Borhom ( ϕ, ψ ) ≤ d hom ( ϕ, ψ ) . By the regularity of b µ ◦ f − ϕ and b µ ◦ f − ψ , forevery B ∈ Bor ( St ( A )) there is A B ∈ A such that B ⊆ A B , b µ (cid:0) f − ϕ [ A B \ B ] (cid:1) < ε/ and b µ (cid:0) f − ψ [ A B \ B ] (cid:1) < ε/ . Thus, for every Borel B ∈ Bor ( St ( A )) we have: b µ (cid:0) f − ϕ [ B ] △ f − ψ [ B ] (cid:1) ≤ b µ (cid:0) f − ϕ [ B ] △ f − ϕ (cid:2) A B (cid:3)(cid:1) + b µ (cid:0) f − ϕ (cid:2) A B (cid:3) △ f − ψ (cid:2) A B (cid:3)(cid:1) + b µ (cid:0) f − ψ (cid:2) A B (cid:3) △ f − ψ [ B ] (cid:1) <ε/ d µ (cid:0) ϕ (cid:0) A B (cid:1) , ψ (cid:0) A B (cid:1)(cid:1) + ε/ ≤ d hom ( ϕ, ψ ) + ε, so d Borhom ( ϕ, ψ ) ≤ d hom ( ϕ, ψ ) + ε . (cid:3) Corollary 4.7.
Let A be a Boolean algebra and ( B , µ ) a metric Boolan algebra. Then:(1) the uniform topology and the uniform Borel topology coincide;(2) a sequence of homomorphisms from A to B converges uniformly if and only ifit converges uniformly Borel. Homomorphisms and measures
By the Maharam theorem for every measure µ on a Boolean algebra A there are acardinal number κ and an injective homomorphism ϕ : A → M κ such that µ = λ κ ◦ ϕ .On the other hand, if ϕ : A → M κ is a homomorphism, then it is plain to check thatthe function λ κ ◦ ϕ : A → R is a (finitely additive) measure. It follows that there isa natural mapping from the family of homomorphisms H ( A , M κ ) into the family ofmeasures on A . This observation allows us to use notions from the theory of measureson Boolean algebras to study H ( A , M κ ) . Proposition 5.1.
Let A be a Boolean algebra and ( B , µ ) a metric algebra. Considerthe space H ( A , B ) with the topology τ and the space P ( St ( A )) with topology τ ′ , bothgiven below. Then, the function F µ : H ( A , B ) → P ( St ( A )) , given by F µ ( ϕ ) = b µ ◦ f − ϕ ,is continuous in each of the following cases:(1) τ is the uniform topology and τ ′ is the norm topology;(2) τ is the pointwise metric topology and τ ′ is the weak* topology.Proof. We will use below the fact that | µ ( C ) − µ ( D ) | ≤ µ ( C △ D ) for every C, D ∈ B .We first prove (1). Using the above inequality, for every A ∈ A we have (cid:12)(cid:12) µ ( ϕ ( A )) − µ ( ψ ( A )) (cid:12)(cid:12) ≤ µ (cid:0) ϕ ( A ) △ ψ ( A ) (cid:1) , and thus, by the definition of the variation metric on P ( St ( A )) , it holds: d var (cid:0) F µ ( ϕ ) , F µ ( ψ ) (cid:1) = sup A,B ∈ A A ∧ B =0 A (cid:16)(cid:12)(cid:12) F µ ( ϕ )( A ) − F µ ( ψ )( A ) (cid:12)(cid:12) + (cid:12)(cid:12) F µ ( ϕ )( B ) − F µ ( ψ )( B ) (cid:12)(cid:12)(cid:17) ≤ A ∈ A (cid:12)(cid:12) F µ ( ϕ )( A ) − F µ ( ψ )( A ) (cid:12)(cid:12) ≤ A ∈ A d µ ( ϕ ( A ) , ψ ( A )) = 2 d hom ( ϕ, ψ ) , which actually shows that F µ is -Lipschitz.To prove (2) let ϕ ∈ H ( A , B ) and fix an element V of the subbase of the weak*topology on P ( St ( A )) of the form V = V ( F µ ( ϕ ) , A, ε ) = (cid:8) ν : (cid:12)(cid:12) F µ ( ϕ )( A ) − ν ( A ) (cid:12)(cid:12) < ε (cid:9) ,where A ∈ A and ε > . Put: U = (cid:8) ψ ∈ H ( A , B ) : d µ ( ϕ ( A ) , ψ ( A )) < ε (cid:9) . Then, U is an open set in H ( A , B ) . For every ψ ∈ U we have: (cid:12)(cid:12) F µ ( ϕ )( A ) − F µ ( ψ )( A ) (cid:12)(cid:12) ≤ d µ (cid:0) ϕ ( A ) , ψ ( A ) (cid:1) < ε, so ϕ ∈ F µ [ U ] ⊆ V , which proves that F µ is continuous. (cid:3) Consequently, the equivalence relation on H ( A , B ) defined by the formula ϕ ∼ ψ ⇐⇒ F µ ( ϕ ) = F µ ( ψ ) , has closed equivalence classes, provided that H ( A , B ) is endowed with any of thetopologies τ mentioned in the proposition. Corollary 5.2.
Let A be a Boolean algebra and ( B , µ ) a metric algebra. Let ϕ n ∈H ( A , B ) , n ∈ ω , and ϕ ∈ H ( A , B ) . The following hold:(1) if ( ϕ n ) converges uniformly to ϕ , then (cid:0)b µ ◦ f − ϕ n (cid:1) converges to b µ ◦ f − ϕ in norm;(2) if ( ϕ n ) converges pointwise Borel metric to ϕ , then (cid:0)b µ ◦ f − ϕ n (cid:1) converges weaklyto b µ ◦ f − ϕ ; N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 11 (3) if ( ϕ n ) converges pointwise metric to ϕ , then (cid:0)b µ ◦ f − ϕ n (cid:1) converges weakly* to b µ ◦ f − ϕ .Proof. (1) and (3) are direct consequences of Proposition 5.1. (2) follows from Propo-sition 3.2. (cid:3) Remark 5.3.
Note here that the converse to Corollary 5.2 may not hold even in thesimplest case when all homomorphisms ϕ n ’s are the same. Indeed, let ( B , µ ) be ametric algebra and let each ϕ n be the identity homomorphism on B . If ϕ ∈ H ( B , B ) is a homomorphism such that µ ( ϕ ( A )) = µ ( A ) and ϕ ( A ) = A for some A ∈ B , thenthe sequence ( ϕ n ) does not converge to ϕ in any of the considered topologies, eventhough µ ( ϕ n ( B )) = µ ( ϕ ( B )) for every n ∈ ω and B ∈ B .We now prove that if a Boolean algebra A has the Grothendieck property, thenevery pointwise metric convergent sequence of homomorphisms from A into a metricalgebra is also pointwise Borel metric convergent. Recall that a sequence ( µ k ) ofRadon measures on a compact Hausdorff space K is uniformly countably additive (also, uniformly exhaustive ) if for every descending sequence ( E n ) of Borel subsetsof K such that T n E n = ∅ and every ε > there is N ∈ ω such that (cid:12)(cid:12) µ k ( E n ) (cid:12)(cid:12) < ε for every n ≥ N and k ∈ ω . Equivalently, there is N ∈ ω such that (cid:12)(cid:12) µ k ( E m ) − µ k ( E n ) (cid:12)(cid:12) < ε for every n, m ≥ N and k ∈ ω (see [Die84, Chapter VII, Theorem 10]for other equivalent definitions). We also say that ( µ n ) is weakly convergent to aRadon measure µ on K if lim n →∞ µ n ( B ) = µ ( B ) for every Borel subset B of K . TheNikodym Convergence Theorem ([Die84, Chapter VII, page 90]) asserts that everyweakly convergent sequence of Radon measures is uniformly countably additive. Proposition 5.4.
Let A be a Boolean algebra and ( B , µ ) be a metric Boolean algebra.Let ( ϕ n ) be a sequence in H ( A , B ) pointwise metric convergent to a homomorphism ϕ ∈ H ( A , B ) . If the sequence (cid:0)b µ ◦ f − ϕ n (cid:1) is uniformly countably additive, then ( ϕ n ) converges pointwise Borel metric to ϕ .Proof. Write simply f n for f ϕ n and f for f ϕ . Fix ε > and a Borel subset B of St ( A ) .Let δ = ε/ . We will show first that b µ (cid:0) f − [ B ] \ f − k [ B ] (cid:1) < δ for almost all k ∈ ω .Since each measure b µ ◦ f − k is (outer) regular on St ( A ) , for each k ∈ ω we can find A k ∈ A such that B ⊆ A k and(1) b µ (cid:0) f − k [ A k \ B ] (cid:1) < δ/ . By taking intersections, we may assume that A k +1 ≤ A k for every k ∈ ω . By theuniform countable additiveness of (cid:0)b µ ◦ f − n (cid:1) , there is k ∈ ω such that(2) b µ (cid:0) f − k [ A l \ A k ] (cid:1) < δ/ for every k > l > k . Since obviously the measure b µ ◦ f − is also countably additive,there is k ≥ k such that(3) b µ (cid:0) f − [ A l \ A k ] (cid:1) < δ/ for every k > l > k . Now, fix l = k + 1 . By the pointwise metric convergence of ( ϕ k ) to ϕ , there is k ≥ l such that(4) b µ (cid:0) f − [ A l ] △ f − k [ A l ] (cid:1) < δ/ for every k > k . Using (3), (4) and (2), for every k > k we have:(5) b µ (cid:0) f − [ A k ] △ f − k [ A k ] (cid:1) ≤ b µ (cid:0) f − [ A k ] △ f − [ A l ] (cid:1) + b µ (cid:0) f − [ A l ] △ f − k [ A l ] (cid:1) + b µ (cid:0) f − k [ A l ] △ f − k [ A k ] (cid:1) < δ . (5) and (1) yield that for every k > k it holds: b µ (cid:0) f − [ B ] \ f − k [ B ] (cid:1) ≤ b µ (cid:0) f − [ A k ] \ f − k [ B ] (cid:1) ≤ b µ (cid:0) f − [ A k ] △ f − k [ B ] (cid:1) ≤ b µ (cid:0) f − [ A k ] △ f − k [ A k ] (cid:1) + b µ (cid:0) f − k [ A k ] △ f − k [ B ] (cid:1) ≤ δ δ δ, so(6) b µ (cid:0) f − [ B ] \ f − k [ B ] (cid:1) < ε/ . Changing B to B c and doing exactly the same computations that led us to (6), weget that there is k ≥ k such that(7) b µ (cid:0) f − k [ B ] \ f − [ B ] (cid:1) = b µ (cid:0) f − [ B c ] \ f − k [ B c ] (cid:1) < ε/ for every k > k . Thus, by (6) and (7), for every k > k we get: b µ (cid:0) f − [ B ] △ f − k [ B ] (cid:1) < ε. This proves that ( ϕ n ) converges pointwise Borel metric to ϕ . (cid:3) Corollary 5.5.
Let A be a Boolean algebra and ( B , µ ) be a metric Boolean algebra.Let ( ϕ n ) be a sequence in H ( A , B ) pointwise metric convergent to a homomorphism ϕ ∈ H ( A , B ) . If A has the Grothendieck property, then ( ϕ n ) converges pointwise Borelmetric to ϕ .Proof. Assume that A has the Grothendieck property. Let ( f k ) and f be the continu-ous functions dual to ( ϕ k ) and ϕ , respectively. Since lim k →∞ b µ (cid:0) f − k [ A ] (cid:1) = b µ (cid:0) f − [ A ] (cid:1) for every A ∈ A , the sequence (cid:0)b µ ◦ f − k (cid:1) is weakly* convergent to b µ ◦ f − (as itis uniformly bounded). The Grothendieck property of A implies that (cid:0)b µ ◦ f − k (cid:1) isweakly convergent to b µ ◦ f − . By the Nikodym Convergence Theorem, the sequenceis uniformly countably additive, so the conclusion follows by Proposition 5.4. (cid:3) Notice here that the measures b µ ◦ f − k considered in the proof of Corollary 5.5 are non-negative , i.e. b µ ◦ f − k ( B ) ≥ for every B ∈ Bor ( St ( A )) . It follows that in theproof we do not need the full Grothendieck property of the algebra A , but only theproperty for non-negative measures, i.e. if a sequence of non-negative measures on A isweakly* convergent, then it is weakly convergent. Such a variant of the Grothendieckproperty was introduced in Koszmider and Shelah [KS13], where it was called thepositive Grothendieck property . Corollary 5.6.
Let A be a Boolean algebra with the positive Grothendieck propertyand ( B , µ ) be a metric Boolean algebra. Let ( ϕ n ) be a sequence in H ( A , B ) pointwisemetric convergent to a homomorphism ϕ ∈ H ( A , B ) . Then, ( ϕ n ) converges pointwiseBorel metric to ϕ . The following weaker form of the σ -completeness was also introduced in [KS13]: aBoolean algebra A is said to have the Weak Subsequential Separation Property ( theWSSP ) if for every sequence ( A n ) of pairwise disjoint elements of A there is an element A ∈ A such that both of the sets (cid:8) n ∈ ω : A n ≤ A (cid:9) and (cid:8) n ∈ ω : A n ∧ A = 0 (cid:9) areinfinite. [KS13, Proposition 2.4] asserts that every Boolean algebra with the WSSPhas the positive Grothendieck property. Note that there exists a Boolean algebrawith the WSSP (and hence with the positive Grothendieck property), but withoutthe Grothendieck property, see [KS13, Proposition 2.5]. N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 13
Recall that by the Eberlein–Šmulian theorem and the Dieudonné–Grothendieckcharacterization of weakly compact sets of Radon measures on compact spaces (see[Die84, Chapter VII, Theorem 14]), a weakly* convergent sequence ( µ n ) of measureson a given compact space K is weakly convergent if and only if there is no antichain ( U k ) of open subsets of K , ε > and subsequence ( µ n k ) such that | µ n k ( U k ) | ≥ ε forevery k ∈ ω . Proposition 5.7.
Let A be a Boolean algebra with the positive Grothendieck property, ( B , µ ) a metric Boolean algebra, and ϕ, ϕ n ∈ H ( A , B ) , n ∈ ω . Let ( A n ) be an antichainin A such that for some ε > and every n ∈ ω we have d µ (cid:0) ϕ n ( A n ) , ϕ ( A n ) (cid:1) ≥ ε . Then, ( ϕ n ) is not pointwise metric convergent to ϕ .Proof. For the sake of contradiction assume that ( ϕ n ) is pointwise metric convergentto ϕ . Then, by Corollary 5.2.(3), the sequence (cid:0)b µ ◦ f − ϕ n (cid:1) is weakly* convergent to b µ ◦ f − ϕ . Since ( A n ) is an antichain in A , the sequence ( ϕ ( A n )) is an antichain in B andthus lim n →∞ µ ( ϕ ( A n )) = 0 . From the assumption that d µ (cid:0) ϕ n ( A n ) , ϕ ( A n ) (cid:1) ≥ ε forevery n ∈ ω it follows that there is N ∈ ω such that µ ( ϕ n ( A n )) ≥ ε/ for every n > N ,so (cid:0)b µ ◦ f − ϕ n (cid:1) ( A n ) ≥ ε/ for every n > N . By the aforementioned characterisation ofweakly convergent sequences of measures on compact spaces, the sequence (cid:0)b µ ◦ f − ϕ n (cid:1) is not weakly convergent to b µ ◦ f − ϕ , and hence, by the positive Grothendieck propertyof A , not weakly* convergent, which is a contradiction. (cid:3) Examples of sequences of homomorphisms
In this section we will present certain examples distinguishing different types ofconvergence. First, we summarize in the following self-explanatory diagram all theimplications which we have already proved or which we are going to prove.uniform pointwise Borel metric pointwise metricpointwise algebraic the positive Grothendieck property Seever’s interpolation property . Of those, (2) is trivial, (1) follows from Corollary 4.7 and (3) is proved in Proposition4.4. No other implication holds in general. (4) holds for Boolean algebras with thepositive Grothendieck property and hence, in particular, for Boolean algebras withthe WSSP (Corollary 5.6). (5) is true for Boolean algebras with the interpolationproperty (or, the property (I)) introduced by Seever (see Section 8). Corollary 4.5implies that (3) may be reversed for B = , but we skipped that in the diagram, sinceit is trivial.Note that it is easy to obtain a sequence which satisfies simultaneously all of theabove types of convergence (e.g. a trivial sequence). Below we present examplesof sequences of homomorphisms satisfying various other combinations of types ofconvergence and witnessing that the above diagram is complete.In what follows we will often abuse the notation identifying equivalence classes(particularly elements of M ) with their representatives. A pointwise algebraically convergent sequence which is not pointwiseBorel metric convergent.
To see that the pointwise algebraic convergence doesnot imply the pointwise Borel metric convergence, consider the Cantor algebra C anda non-trivial sequence ( x n ) of points in the Cantor space St ( C ) = 2 ω convergent tosome x ∈ ω . For each n ∈ ω define ϕ n ∈ H ( C , ) by the condition: ϕ n ( A ) = 1 iff x n ∈ A , where A ∈ C . Similarly, define ϕ ∈ H ( C , ) by the condition: ϕ ( A ) = 1 iff x ∈ A , where A ∈ C . Since lim n →∞ x n = x in St ( C ) , ( ϕ n ) is pointwise algebraicallyconvergent. On the other hand, ( ϕ n ) does not converge pointwise Borel metric to ϕ ,since for the Borel set { x } and every n ∈ ω we have: b λ (cid:0) f − ϕ n (cid:2) { x } (cid:3) △ f − ϕ (cid:2) { x } (cid:3)(cid:1) = λ (0 △
1) = 1 . Of course, in the above example instead of C we may use any Boolean algebra suchthat its Stone space contains a non-trivial convergent sequence.6.2. A uniformly convergent sequence which is not pointwise algebraicallyconvergent.
Here we will present an example of a sequence of homomorphisms in H ( C , M ) which converges uniformly but not pointwise algebraically.Let { s n : n ∈ ω } be any enumeration of <ω . For x ∈ ω define x ∈ ω by x ( k ) = ( − x ( k ) , if k = 0 ,x ( k ) , otherwise,so x is just x but with the -th bit flipped. For s ∈ <ω and x ∈ ω we will saythat x agrees with s , if x ( k + 1) = s ( k ) for each k < | s | (so, in other words, s is asegment of x starting with the -st bit).Now for n ∈ ω define g n : 2 ω → ω by g n ( x ) = ( x, if x agrees with s n ,x, otherwise.It is easy to check that ψ n : C → M defined by ψ n ( A ) = g n [ A ] = g − n [ A ] is ahomomorphism. Let ψ be the natural embedding of C into M , i.e. ψ ( A ) = A forevery A ∈ C . (Note that here we treat C as the algebra of clopen subsets of ω andwriting g n [ A ] we mean the appropriate equivalence class, that is, an element of M .) Proposition 6.1. ( ψ n ) converges uniformly to ψ but it does not converge pointwisealgebraically.Proof. For each s ∈ <ω let A s = { x ∈ ω : x agrees with s } . Clearly, λ ( A s ) = 1 / | s | and so lim n →∞ λ ( A s n ) = 0 . But ψ n coincides with ψ on ω \ A s n and so ( ψ n ) convergesuniformly to ψ .Let A = { x ∈ ω : x (0) = 0 } . Fix x ∈ ω and n ∈ ω . If x ∈ A , then there is m > n such that x agrees with s m and so x g m [ A ] . Hence, A ∩ T m>n g m [ A ] = ∅ . If x A ,then there is m > n such that x does not agree with s m and so x g m [ A ] . Hence, A c ∩ T m>n g m [ A ] = ∅ , and so T m>n g m [ A ] = ∅ . Since n was arbitrary, we get that S n T m>n g m [ A ] = ∅ and hence W n V m>n ψ n ( A ) = 0 = A = ψ ( A ) , which implies that ( ϕ n ) does not converge pointwise algebraically to ψ . (cid:3) Note that since the pointwise algebraic convergence implies the pointwise metricconvergence, it is not possible that ( ψ n ) converges pointwise algebraically to a homo-morphism other than ψ . N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 15
A pointwise metric convergent sequence which is neither pointwiseBorel metric convergent nor pointwise algebraically convergent.
Combiningmethods used in Sections 6.1 and 6.2, we may easily obtain an example of a sequenceof homomorphisms in H ( C , M ) which is only pointwise metric convergent.For each n ∈ ω , let ϕ n : C → be defined as in Section 6.1 and let ψ n : C → M be defined as in Section 6.2. Define ρ n : C → × M by the formula ρ n ( A ) =( ϕ n ( A ) , ψ n ( A )) for every A ∈ C .By the same arguments as in Sections 6.1 and 6.2, the sequence ( ρ n ) is point-wise metric convergent to the homomorphism ρ = ( ϕ, ψ ) , yet, again by the samearguments, it is not pointwise Borel metric convergent nor pointwise algebraicallyconvergent. Of course, × M can be embedded into M , so ( ρ n ) can be treated as asequence in H ( C , M ) .6.4. A pointwise Borel metric convergent sequence which is neither uni-formly convergent nor pointwise algebraically convergent.
For every n ∈ ω define f lip n : 2 ω → ω by f lip n ( x ) = x + χ { n } , where + is pointwise modulo (so, in short, f lip n flips the n -th coordinate of x ). In the analogous way we de-fine f lip n also on <ω \
There is a closed subset B of ω such that λ ( B ) > / and foreach x ∈ B we have f lip n ( x ) / ∈ B for infinitely many n ’s. Perhaps quite unexpectedly, to prove it we will use several notions from codingtheory. For n ∈ ω the function ρ n : { , } n → ω defined by ρ n ( x, y ) = (cid:12)(cid:12) { i ≤ n : x ( i ) = y ( i ) } (cid:12)(cid:12) is clearly a metric (called the Hamming metric ). We will call a subset A ⊆ { , } n acode if for every x ∈ { , } n there is y ∈ A such that ρ n ( x, y ) ≤ . In coding theorythis property is one of the conditions defining so-called 1-error correcting codes.In what follows we will use the fact that there is a perfect sphere packing in { , } n for each n of the form n = 2 m − , m ∈ ω , i.e. { , } n is a disjoint union of theclosed balls of radius with respect to the Hamming metric (see [Ham50]). The setof centers of those balls is a code, which we will call a perfect code (in the codingtheory terminology this is the same as a perfect 1-error-correcting Hamming code).There is a rich literature concerning Hamming codes and it is quite easy to learn how to find perfect codes. It seems, however, that it is not so easy to find a proof why those algorithms work, so we sketch here the argument, using the algorithm forcreating perfect codes presented e.g. in [AK92, Section 2.5]. Proposition 6.3. If n = 2 m − , for m ∈ N , then { , } n is a disjoint union of theclosed balls of radius with respect to the Hamming metric.Proof. Let n = 2 m − . We endow the set { , } n with the algebraic structure and seeit as F n (i.e. n -dimensional vector space over F , the field consisting of two elements).Let H be a matrix created by putting all the non-zero elements of F m as its columns.Then H is an ( n × m ) -matrix. Define C = (cid:8) v ∈ F n : Hv = 0 F n (cid:9) . In other words, v ∈ C if and only if P l By (2) α > and so, using Proposition 6.4, we may find k so that a k < α . Let C bea code in { , } k such that | C | = a k k . Define C n +1 in the following way: C n +1 = (cid:8) t ⌢ x : t ∈ C n , x ∈ { , } k (cid:9) ∪ (cid:8) t ⌢ c : t ∈ { , } d n \ C n , c ∈ C (cid:9) . Thus, C n +1 consists of two parts: all the possible extensions of elements of C n to { , } d n + k and all the extensions of the rest of { , } d n by elements of C . Put d n +1 = d n + k . We claim that C n +1 and d n +1 satisfy all the desired properties. Indeed, (1)and (3) are clear, and concerning (2) we have: | C n +1 | = 2 k | C n | + (2 d n − | C n | ) · a k k , so | C n +1 | d n +1 = | C n +1 | d n + k = | C n | d n + (2 d n − | C n | ) · a k d n << | C n | d n + (2 d n − | C n | )2 d n · (2 d n − − | C n | )(2 d n − | C n | ) = 2 d n − d n = 12 , hence | C n +1 | < d n +1 − .Finally, to check (4) let s ∈ { , } d n +1 \ C n +1 and t = s ↾ d n . If t ∈ C n , then f lip m ( s ) ∈ C n +1 for every d n ≤ m < d n +1 . If t C n , then, as C is a code, there is c ∈ C such that ρ d n +1 ( t ⌢ c, s ) ≤ , which means that there is d n ≤ m < d n +1 suchthat f lip m ( s ) = t ⌢ c and so f lip m ( s ) ∈ C n +1 .Let B n = 2 ω \ S c ∈ C n [ c ] . By (3), the sequence ( B n ) is ⊆ -decreasing. Let B = T n B n .Because of (2) we have that λ ( B n ) > / for each n and so λ ( B ) ≥ / .Let x ∈ ω . If x / ∈ B , then c = x ↾ d n ∈ C n for some n . Then f lip m ( x ) ∈ [ c ] forevery m > d n and so f lip m ( x ) / ∈ B for every m > d n . Now, assume that x ∈ B , so x ↾ d n C n for any n ∈ ω . Fix N ∈ ω . Find n such that d n > N and let s = x ↾ d n +1 ,so that s ∈ { , } d n +1 \ C n +1 . By (4), there is m ≥ d n such that f lip m ( s ) ∈ C n +1 .This means that f lip m ( x ) / ∈ B n +1 and so f lip m ( x ) / ∈ B . Since N was arbitrary, weare done. (cid:3) Corollary 6.5. ( ϕ n ) does not converge pointwise algebraically.Proof. Let B be a set given by Proposition 6.2. For each m we have \ n>m f lip n [ B ] = ∅ . Indeed, if x ∈ T n>m f lip n [ B ] , then for every n > m there is y n ∈ B such that x = f lip n ( y n ) . But this means that f lip n ( x ) = y n ∈ B for each n > m , which is acontradiction with the property of B promised by Proposition 6.2.It follows that W m V n>m ϕ n ( B ) = 0 , but B = 0 and so ( ϕ n ) does not convergepointwise algebraically to the identity ψ . (cid:3) Remark 6.6. Notice that in fact in Proposition 6.2 we can ask for B of arbitrarilylarge measure (smaller than )—it is enough to adjust the definition of α in the proof. Remark 6.7. Using the Vitali equivalence relation, it is easy to construct a non-measurable set A such that λ ∗ ( A ) = 1 with even stronger property: for each x ∈ A and every n ∈ ω we have f lip n ( x ) / ∈ A . Using this set is probably the easiest methodof showing the existence of a infinite game with perfect information which is notdetermined (see e.g. [KN14]). It seems however that to obtain a Borel set, we have to be much more careful and we do not know any way to do this without using Hammingcodes.6.5. A sequence which is pointwise Borel metric convergent and pointwisealgebraically convergent but not uniformly convergent. The above examplemay be adapted to obtain a sequence which is pointwise Borel metric convergent andpointwise algebraically convergent but not uniformly convergent.Let ( ϕ n ) be the sequence defined in Section 6.4. Define ψ n : C → M by ψ n = ϕ n ↾ C .Here, we treat C as a subalgebra of M , identifying clopen subsets of ω with itsequivalence relations (i.e. the elements of M ). By the same argument which showedthat ( ϕ n ) was pointwise metric convergent to the identity on M , ( ψ n ) is pointwise Borel metric convergent to the identity on C . Since the sets X n ’s from Section 6.4 areclopen in ω , ( ψ n ) is not uniformly convergent. It is however pointwise algebraicallyconvergent, since f lip n [ C ] is eventually constant for C being a clopen subset of ω .In Section 8 we prove however that if a Boolean algebra A is somewhat more com-plicated than C , e.g. A is σ -complete, then there is no similar example in H ( A , M κ ) .7. Homomorphisms and forcing names for ultrafilters Fix, once and for all, an infinite cardinal κ and suppose that we force with thenotion M κ over a model V of set theory. If A is a Boolean algebra in V and G isan M κ -generic filter over V , then we can consider A as an element of V [ G ] , however,there might be new ultrafilters on A in V [ G ] . It appears nevertheless that to each M κ -name ˙ U for an ultrafilter, i.e. to such ˙ U that (cid:13) M κ “ ˙ U is an ultrafilter on A ”, wecan assign a homomorphism from A to M κ in V in a quite natural way. Namely, if ˙ U is such an M κ -name, then the function ϕ ˙ U : A → M κ defined fo every A ∈ A (recallthat A ∈ V ) by the formula ϕ ˙ U ( A ) = J A ∈ ˙ U K is a Boolean homomorphism belonging to V . (We skip writting the symbol ˇ overcanonical names for elements of V .)Now fix a homomorphism ϕ : A → M κ in V and define an M κ -name τ ϕ for a familyof subsets A by τ ϕ = (cid:8) h A, ϕ ( A ) i : A ∈ A (cid:9) . Then, (cid:13) M κ “ τ ϕ is an ultrafilter on A ”; in fact, we actually get that (cid:13) M κ τ ϕ = ϕ − [Γ] ,where Γ is the canonical M κ -name for an M κ -generic filter over V . Lemma 7.1. For every M κ -name ˙ U for an ultrafilter on a Boolean algebra A and forevery homomorphism ϕ : A → M κ it holds ϕ ( τ ϕ ) = ϕ and (cid:13) M κ τ ( ϕ ˙ U ) = ˙ U . In this way we obtain a bijective correspondence between elements of H ( A , M κ ) andultrafilters on A in the forcing extension by M κ . One of the advantages of such anapproach is that we may use properties of measures associated to homomorphisms (asin Section 4) to study M κ -names for ultrafilters on old Boolean algebras (in particular,to study ultrafilters on the Cantor algebra, i.e. reals). Example 7.2. Suppose that ϕ : A → M κ is such that the measure λ κ ◦ ϕ is a pointmass measure δ V for some V ∈ St ( A ) ∩ V . Then, (cid:13) M κ τ ϕ = V .A study of the interplay between the properties of M κ -names for ultrafilters andmeasures is a subject of forthcoming paper [BNC]. N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 19 In what follows ( ϕ n ) and ϕ will denote (a sequence of) elements of H ( A , M κ ) associated—as described above—to (a sequence of) M κ -names ( ˙ U n ) and ˙ U for ultra-filters on a fixed Boolean algebra A ∈ V , respectively. Note that by Lemma 7.1, ithas no meaning whether we associate homomorphisms in H ( A , M κ ) to M κ -names, or M κ -names to homomorphisms in H ( A , M κ ) .Recall here that for every sequence ( φ n ) of formulae the following equalities hold: J ∃ ∞ n : φ n K = J ∀ n ∃ m > n φ n K = ^ n _ m>n J φ n K and J ∀ ∞ n : φ n K = J ∃ n ∀ m > n φ n K = _ n ^ m>n J φ n K ; we will use them frequently.7.1. Pointwise algebraic convergence. There is a strong connection between thepointwise algebraic convergence of sequences of homomorphisms from a Boolean al-gebra A into M κ in the ground model V and the convergence of the sequences ofcorresponding ultrafilters in the Stone space St ( A ) in the M κ -generic extension V [ G ] . Proposition 7.3. The following conditions are equivalent: • (cid:13) M κ “( ˙ U n ) converges to ˙ U ”, • ( ϕ n ) converges to ϕ pointwise algebraically.Proof. Suppose that it is not true that (cid:13) M κ “( ˙ U n ) converges to ˙ U ”. It means thatthere is a non-zero p ∈ M κ which forces the opposite, i.e. that there is A ∈ A suchthat p (cid:13) ∃ ∞ n A ∈ ˙ U \ ˙ U n . It follows that p ≤ ^ m _ n>m J A ∈ ˙ U \ ˙ U n K , and hence ^ m _ n>m (cid:0) ϕ ( A ) \ ϕ n ( A ) (cid:1) = 0 . Thus, ( ϕ n ( A )) does not converge to ϕ ( A ) and hence ( ϕ n ) does not converge pointwisealgebraically to ϕ .If ( ϕ n ( A )) does not converge algebraically to ϕ ( A ) for some A ∈ A , then p = ϕ ( A ) \ (cid:0) ^ m _ n>m ϕ n ( A ) (cid:1) = 0 or q = (cid:0) ^ m _ n>m ϕ n ( A ) (cid:1) \ ϕ ( A ) = 0 . Assume first that the former case holds, i.e. p = 0 . We have: p = ϕ ( A ) ∧ (cid:0) _ m ^ n>m ϕ n ( A c ) (cid:1) , so p (cid:13) A ∈ ˙ U and p (cid:13) ∃ ∞ n A c ∈ ˙ U n . It follows that p (cid:13) “( ˙ U n ) does not converge to ˙ U ”.We proceed similarly in the latter case, i.e. when q = ∅ . (cid:3) Uniform convergence. In this section we will show that the notion of uniformconvergence of homomorphisms is connected to trivial convergence of ultrafilters inthe random extension. Proposition 7.4. If (cid:13) M κ ∀ ∞ n ∈ ω ˙ U n = ˙ U , then ( ϕ n ) converges uniformly to ϕ .Proof. Assume that ( ϕ n ) does not converge uniformly to ϕ . Then, without loss ofgenerality, there is ε > such that for each n ∈ ω there is A n ∈ A such that λ (cid:0) ϕ ( A n ) \ ϕ n ( A n ) (cid:1) > ε . For every n ∈ ω put p n = ϕ ( A n ) \ ϕ n ( A n ) . Since λ (cid:0) W n>m p n (cid:1) > ε forevery m ∈ ω , for p = V m W n>m p n we have λ ( p ) ≥ ε and thus p = 0 .Let G be an M κ -generic filter over V containing p . We work in V . There is q ∈ G below p such that q (cid:13) M κ ∀ ∞ n ∈ ω ˙ U n = ˙ U , and hence there is r ∈ G below q and m ∈ ω such that for every n > m we have r (cid:13) ˙ U n = ˙ U . It follows immediately that p n ⊥ r for every n > m and thus p ⊥ r , a contradiction, since r ≤ p . (cid:3) It appears that a fact which is in a sense converse to Proposition 7.4 also holds andconstitutes actually the main result of this section (Theorem 7.5). Note here that inthe proof of Proposition 7.4, as well as of Proposition 7.3, we actually did not usethat the Boolean algebra M κ carries a strictly positive measure and thus those resultsremain in fact true for any σ -complete Boolean algebra B in place of M κ (and hence,of course, for any reasonable notion of forcing). Theorem 7.5. If ( ϕ n ) converges uniformly to ϕ , then for almost all n there is p n ∈ M κ such that p n (cid:13) ˙ U n = ˙ U . We will prove this theorem in a series of lemmas and propositions. One of them,Theorem 7.6, is interesting on its own sake. Theorem 7.6. Let A be an atomless Boolean algebra in V . If ˙ U and ˙ V are M κ -namesfor ultrafilters on A such that (cid:13) M κ ˙ U 6 = ˙ V , then for every ε > there is p ∈ M κ and C ∈ A such that λ ( p ) > / − ε and p (cid:13) C ∈ ˙ U △ ˙ V . Theorem 7.6 requires a brief comment. Assume that A is a Boolean algebra and ˙ U and ˙ V are M κ -names for ultrafilters on A such that (cid:13) M κ ˙ U 6 = ˙ V . Since M κ is a cccforcing, there is a maximal antichain ( p n ) in M κ and a sequence ( A n ) of elements of A such that p n (cid:13) A n ∈ ˙ U \ ˙ V . The problem with this general statement is that a priori we do not have any control on the measures of p n ’s—it might as lief happen that allof them are very small in terms of the measure. Theorem 7.6 shows however that wecan always find one “large” condition in M κ which distinguishes the ultrafilters.The proof of Theorem 7.6 requires some auxiliary lemmas. Proposition 7.7. Assume that ( X, Σ , µ ) is a measure space with a non-negative (notnecessarily probability) measure µ , K is a compact space, and f, g : X → K are simplemeasurable functions. If f = g µ -almost everywhere, then there exists L ∈ Σ suchthat µ ( L ) ≥ µ ( X ) / and f [ L ] ∩ g [ L ] = ∅ .Proof. Let P be a finite partition of X into measurable pieces such that for every A ∈ P the functions f and g are constant on A and µ ( A ) > . Define the functions f P , g P : P → K as follows: f P ( A ) = x if and only if f [ A ] = { x } , and similarly g P ( A ) = x if and only if g [ A ] = { x } . N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 21 For every A ∈ P we have f P ( A ) = g P ( A ) (since µ ( A ) > ). Put H = f [ X ] ∩ g [ X ] .Define an auxiliary measure ν on P ( H × H ) as follows: ν (cid:0) { ( x, y ) } (cid:1) = X (cid:8) µ ( A ) : f P ( A ) = x, g P ( A ) = y, A ∈ P (cid:9) for every ( x, y ) ∈ H × H . It follows that ν (cid:0) { ( x, x ) } (cid:1) = 0 for every x ∈ H .For every N ⊆ H let N = ( N × N ) ∪ ( N c × N c ) and N = ( N × N c ) ∪ ( N c × N ) . It follows that for every ( x, y ) ∈ H × H with x = y we have: (cid:12)(cid:12) { N ⊆ H : ( x, y ) ∈ N } (cid:12)(cid:12) = (cid:12)(cid:12) { N ⊆ H : ( x, y ) ∈ N } (cid:12)(cid:12) , and so X N ∈P ( H ) ν ( N ) = X N ∈P ( H ) ν ( N ) . There exists N ⊆ H such that ν ( N ) ≥ ν ( N ) . Since N ∪ N = H × H , ν ( N ) ≥ ν ( H × H ) / and hence either ν ( N × N c ) ≥ ν ( H × H ) / or ν ( N c × N ) ≥ ν ( H × H ) / .Without loss of generality we may assume that ν ( N × N c ) ≥ ν ( H × H ) / as thisassumption makes no difference for the following arguments.For the set L ′ ⊆ X defined as follows: L ′ = [ (cid:8) A ∈ P : f P ( A ) ∈ N, g P ( A ) ∈ N c (cid:9) , we have: µ ( L ′ ) = X (cid:8) µ ( A ) : f P ( A ) ∈ N, g P ( A ) ∈ N c , A ∈ P (cid:9) = ν ( N × N c ) ≥≥ ν ( H × H ) / X (cid:8) µ ( A ) : f P ( A ) ∈ H, g P ( A ) ∈ H, A ∈ P (cid:9) . Note that for every x ∈ L ′ we have f ( x ) ∈ N and g ( x ) ∈ N c , so neither f ( x ) ∈ g [ L ′ ] ,nor g ( x ) ∈ f [ L ′ ] . Finally, for the set L ⊆ X given by the formula: L = L ′ ∪ [ (cid:8) A ∈ P : f P ( A ) H or g P ( A ) H (cid:9) , we also have that f [ L ] ∩ g [ L ] = ∅ as well as it holds: µ ( L ) = µ ( L ′ ) + X (cid:8) µ ( A ) : f P ( A ) H or g P ( A ) H (cid:9) ≥≥ X (cid:8) µ ( A ) : f P ( A ) ∈ H, g P ( A ) ∈ H (cid:9) + 14 X (cid:8) µ ( A ) : f P ( A ) H or g P ( A ) H (cid:9) == 14 X (cid:8) µ ( A ) : A ∈ P (cid:9) = µ ( X ) / . (cid:3) Note that in the above proposition we may in fact assume that K is finite. Lemma 7.8. Assume that ( X, Σ , µ ) is a probability space, K is a closed subset of ω ,and f, g : X → K are measurable and f = g µ -almost everywhere. Then, for every ε > there is L ∈ Σ such that µ ( L ) > / − ε and f [ L ] ∩ g [ L ] = ∅ . Proof. For every s ∈ <ω let s ∈ ω be the leftmost branch with root s (think of s as of a fixed representative of the clopen { x ∈ ω : s ⊆ x } ). For n ∈ ω define thesimple measurable functions f n , g n : X → K by putting f n ( x ) = s and g n ( x ) = t ,where s = f ( x ) ↾ n and t = g ( x ) ↾ n .For every n ∈ ω put A n = (cid:8) x ∈ X : f n ( x ) = g n ( x ) (cid:9) and note that A n +1 ⊆ A n . Itfollows that µ (cid:0) A n (cid:1) < ε for some n ∈ ω . Indeed, if µ ( A n ) ≥ ε for every n ∈ ω , then µ ( T n A n ) ≥ ε , and since f ( x ) = g ( x ) for every x ∈ T n A n , we get a contradictionwith the assumption that f = g µ -almost everywhere.Let X = X \ A n , so µ ( X ) > − ε and for every x ∈ X we have f n ( x ) = g n ( x ) . ByLemma 7.7 there is a measurable subset L ⊆ X such that µ ( L ) ≥ / µ ( X ) > / − ε and f n [ L ] ∩ g n [ L ] = ∅ . It follows that for every x, x ′ ∈ L we have f n ( x ) = g n ( x ′ ) andhence f ( x ) = g ( x ′ ) , so L is as desired. (cid:3) Proposition 7.9. Let A be a countable Boolean algebra. Assume that ϕ : A → M κ and ψ : A → M κ are such homomorphisms that W (cid:8) ϕ ( A ) △ ψ ( A ) : A ∈ A (cid:9) = 1 . Then,for each ε > there is C ∈ A such that λ κ (cid:0) ϕ ( C ) △ ψ ( C ) (cid:1) > / − ε .Proof. Let X and Y be the Stone spaces of M κ and A , respectively. In what follows,we will treat λ κ as a Radon measure on X . The functions f ϕ , f ψ : X → Y are λ κ -measurable. We claim that f ϕ = f ψ λ κ -almost everywhere. If not, then there is a Borelsubset B ⊆ X of positive measure such that f ϕ ↾ B = f ψ ↾ B . By the regularity of λ κ ,there is a compact subset K of X such that K ⊆ B and λ κ ( K ) > . Recall that λ κ is a normal measure, i.e. for every nowhere dense subset Z of X we have λ κ ( Z ) = 0 , so K cannot be nowhere dense and thus there is U ∈ M κ such that U ⊆ K and λ κ ( U ) > .But for every A ∈ A we have ( ϕ ( A ) \ ψ ( A )) ∩ U = 0 (since otherwise for each x ∈ ( ϕ ( A ) \ ψ ( A )) ∩ U we would have f ϕ ( x ) ∈ A and f ψ ( x ) / ∈ A and so f ϕ ( x ) = f ψ ( x ) despite the fact that x ∈ B ) and ( ψ ( A ) \ ϕ ( A )) ∩ U = 0 (by a similar argument). Thisis a contradiction with the assumption that W (cid:8) ϕ ( A ) △ ψ ( A ) : A ∈ A (cid:9) = 1 .Now fix ε > and use Lemma 7.8 to find a measurable subset L ⊆ X such that λ κ ( L ) > / − ε and f ϕ [ L ] ∩ f ψ [ L ] = ∅ (notice that Y may be treated as a closedsubset of ω ). Using the regularity of λ κ , we may find a compact subset M of X suchthat M ⊆ L and λ κ ( M ) > / − ε . By the normality of λ κ , the boundary of M hasmeasure , so there is V ∈ M κ such that V ⊆ M and λ κ ( V ) > / − ε . Let C be aclopen in Y separating f ϕ [ V ] and f ψ [ V ] . Then, V ⊆ ϕ ( C ) but V ∩ ψ ( C ) = ∅ , andhence λ ( ϕ ( C ) \ ψ ( C )) > / − ε . (cid:3) Proof of Theorem 7.6 . Since M κ is a ccc poset, there are a sequence ( A n ) in A anda maximal antichain ( p n ) in M κ such that p n ≤ ϕ ˙ U ( A n ) △ ϕ ˙ V ( A n ) for every n ∈ ω . Itfollows that _ (cid:8) ϕ ˙ U ( A n ) △ ϕ ˙ V ( A n ) : n ∈ ω (cid:9) = 1 . Let B be the subalgebra of A generated by the set { A n : n ∈ ω } . By Proposition7.9, there is C in B (and hence in A ) such that λ κ (cid:0) ϕ ˙ U ( C ) △ ϕ ˙ V ( C ) (cid:1) > / − ε . Put p = ϕ ˙ U ( C ) △ ϕ ˙ V ( C ) to finish the proof. (cid:3) Finally, we are in the position to prove the main theorem of this section. Proof of Theorem 7.5 . Since ( ϕ n ) converges uniformly to ϕ , there is m ∈ ω suchthat for each n > m and A ∈ A we have λ κ (cid:0) ϕ n ( A ) △ ϕ ( A ) (cid:1) < / . We get that thereis m ∈ ω such that for each n > m there are no p ∈ M κ and no A ∈ A such that N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 23 p ≤ ϕ n ( A ) △ ϕ ( A ) and λ κ ( p ) ≥ / , which, by Theorem 7.6, means that for every n > m we have a non-zero p n ∈ M κ such that p n (cid:13) M κ ˙ U n = ˙ U . (cid:3) Surprisingly, in general we cannot hope to find anything better than the constant / in Proposition 7.7 and in Theorem 7.6. Lemma 7.10. For each non-zero even n ∈ ω put X n = { , , . . . , n · ( n − } andlet µ n be such that µ n ( { x } ) = 1 / | X n | for each x ∈ X n . Let f n , g n : X n → ω be suchfunctions that for every distinct a, b ∈ { , . . . , n } there is x ∈ X n for which we have f n ( x ) = a and g n ( x ) = b . Then, for each subset L of X n such that f n [ L ] ∩ g n [ L ] = ∅ we have: µ n ( L ) ≤ 14 + 116 n − . Proof. Let L ⊆ X n be a set such that f n [ L ] ∩ g n [ L ] = ∅ . If | f n [ L ] | = k for some ≤ k ≤ n , then | g n [ L ] | ≤ n − k . Since each pair ( a, b ) ∈ { , . . . , n } , a = b ,occurs exactly once in the set f n [ X n ] × g n [ X n ] , L may have cardinality at most k · ( n − k ) . The only maximum of the function ρ n : R → R given by the formula ρ n ( x ) = x · ( n − x ) occurs at x = n/ and is equal to n / . It follows that themaximal possible cardinality of L is also n / and thus: µ n ( L ) ≤ n · n ( n − 1) = 14 − n = 14 + 116 n − . (cid:3) Corollary 7.11. Let A be an infinite Boolean algebra. Then, for every ε > thereare M κ -names ˙ U and ˙ V for ultrafilters on A having the following properties: • (cid:13) M κ ˙ U 6 = ˙ V , and • for every p ∈ M κ for which there is A ∈ A such that p (cid:13) A ∈ ˙ U \ ˙ V we have λ κ ( p ) ≤ / ε .Proof. Let ε > and fix n such that n − < ε . Let ( U i ) i 14 + ε > 14 + 116 n − . Thus, by Lemma 7.10 we have f [ L ] ∩ g [ L ] = ∅ and so there are k, l ∈ L such that f ( k ) = g ( l ) = i for some i < n . Then P k ∧ p and P l ∧ p are non-zero and P k ∧ p ≤ F i (cid:13) ˙ U = U i but also P l ∧ p ≤ G i (cid:13) ˙ V = U i . Hence, there is no A ∈ A such that p (cid:13) A ∈ ˙ U \ ˙ V . (cid:3) Remark 7.12. The core argument of Theorem 7.6 is contained in Proposition 7.7,which is a statement in finite combinatorics. It can be formulated in a popular way asfollows. Adam and Eve are painting a picket fence between their properties. The fenceconsists of n many rails. Each rail has two sides—Adam’s and Eve’s—and those sideshave to be painted in such a way that Adam’s side has different colour than Eve’s.Proposition 7.7 says that no matter how many colours they use, there is always a set B of at least n/ many rails with the following property: the set of Adam’s coloursused in B is disjoint with the set of Eve’s colours used in B . (The translation of theabove into the terms of Proposition 7.7 is as follows: X is the set of rails, K is theset of colors, µ is the counting measure, and the functions f and g assign colors toAdam’s and Eve’s sides of the fence.)8. The Efimov problem We now use results from the previous section to obtain a characterization of thoseBoolean algebras whose Stone spaces do not contain any non-trivial convergent se-quences in random extensions.Recall that a Boolean algebra A has Seever’s interpolation property (or, property(I) ) if for every sequences ( A n ) and ( B n ) in A such that A n ≤ B m for every n, m ∈ ω there is C ∈ A such that A n ≤ C ≤ B m for every n, m ∈ ω (see [See68]). It isimmediate that a Boolean algebra A has the interpolation property if and only if itsStone space St ( A ) is an F-space, so e.g. P ( ω ) /F in has the interpolation property aswell as all σ -complete Boolean algebras do. The result of Dow and Fremlin mentionedin Introduction yields thus the following corollary. Theorem 8.1. [DF07, Corollary 2.3] Let A be a Boolean algebra in V . If A has theinterpolation property in V , then in V M κ the space St( A ) does not have non-trivialconvergent sequences. Theorem 8.1 together with results obtained above allows us to bring down the ques-tion about convergence of ultrafilters in the random extensions to the question aboutconvergence of homomorphisms. Note that if the Stone space St ( A ) of a Booleanalgebra A contains a non-trivial convergent sequence in V , then St ( A ) contains sucha sequence in V M κ , too. The following theorem implies thus e.g. that every countableBoolean algebra admits a sequence of homomorphisms into M κ which is pointwisealgebraically convergent but not uniformly convergent (since St ( A ) is metrizable; cf.the examples in Sections 6.1 and 6.5). Theorem 8.2. Let A be a Boolean algebra in V . The following are equivalent:(1) (cid:13) M κ St ( A ) does not have non-trivial convergent sequences;(2) every pointwise algebraically convergent sequence of homomorphisms from A into M κ converges uniformly. N SEQUENCES OF HOMOMORPHISMS INTO MEASURE ALGEBRAS 25 Proof. (1) = ⇒ (2). Suppose that ( ϕ n ) in H ( A , M κ ) converges pointwise algebraicallyto some homomorphism ϕ . Then, by Proposition 7.3, (cid:13) M κ “ τ ϕ n converges to τ ϕ ”. Bythe assumption, (cid:13) M κ ∀ ∞ n ∈ ω τ ϕ n = τ ϕ , and so, by Proposition 7.4, ( ϕ n ) convergesuniformly to ϕ .(2) = ⇒ (1). Suppose now that there exist p ∈ M κ , and M κ -names ( ˙ U n ) , ˙ U forultrafilters on A such that p (cid:13) M κ “( ˙ U n ) converges to ˙ U and ∀ n ˙ U n = ˙ U ”. Since M κ isisomorphic to the restricted forcing p ∧ M κ = (cid:8) q ∈ M κ : q ≤ p (cid:9) and hence M κ -genericextensions of V are the same as ( p ∧ M κ ) -generic extensions, we may assume that p = 1 .By Proposition 7.3, the sequence ( ϕ ˙ U n ) converges pointwise algebraically to ϕ ˙ U and hence, by the assumption, uniformly. Theorem 7.5 implies that there is n and anon-zero p ∈ M κ such that p (cid:13) ˙ U n = ˙ U , a contradiction. It follows that there are nonon-trivial convergent sequences in St ( A ) ∩ V M κ . (cid:3) Since a compact space containing a copy of βω must necessarily have weight atleast ω and the forcing M κ preserves cardinals, we immediately get the followingcorollary. Corollary 8.3. Assume that A ∈ V is a Boolean algebra of size < κ and suchthat in V every pointwise algebraically convergent sequence in H ( A , M κ ) is uniformlyconvergent. Then, in V M κ , the Stone space of A is an Efimov space. The theorem of Dow and Fremlin yields the following corollary. Corollary 8.4. If A has the interpolation property, then every pointwise algebraicallyconvergent sequence of homomorphisms in H ( A , M κ ) converges uniformly. In partic-ular, this holds if A is σ -complete or A = P ( ω ) /F in . Recall that in Section 6.5 we presented an example of a pointwise algebraic con-vergent sequence of homomorphisms from the Cantor algebra C to M which is notuniformly convergent—this example, as contradicting the conclusion of Corollary 8.4,shows that a Boolean algebra A in the corollary cannot be to simple.9. Open questions and problems Fix two Boolean algebras A and B and an infinite cardinal number κ . In thisfinal section we briefly list several open questions and problems concerning the space H ( A , B ) . Question 9.1. What topological properties does the space H ( A , B ) have when en-dowed with the topologies defined in Section 4? In particular, what properties doesthe space H ( M κ , M κ ) have?The following two questions seem natural in the context of Proposition 5.1 andCorollary 5.2. Question 9.2. Is the function F µ from Proposition 5.1 continuous if the spaces H ( A , B ) and P ( St ( A )) are endowed with the pointwise Borel metric topology andweak topology, respectively? Problem 9.3. Let A be a Boolean algebra and κ a cardinal. Is there a naturalconvergence on the space P ( St ( M κ )) which corresponds to the pointwise algebraicconvergence in H ( A , M κ ) in the sense of Corollary 5.2? The following question concerns possible extensions of the diagram from the be-ginning of Section 6. Question 9.4. What algebraic or structural properties of the Boolean algebras A and B imply that a sequence in H ( A , B ) convergent in one topology is also convergentin some other one?In Section 6 we have presented several examples of sequences of homomorphismswitnessing that the convergence in one topology may not be sufficient to imply theconvergence in some other one. We believe that some of those examples may begeneralized to work in more generic situations. Problem 9.5. Provide a general scheme for constructing a definable sequence ofhomomorphisms in H ( A , B ) such that it is convergent in one topology but not in theother one, or a general scheme for constructing a Borel set or a sequence of Borelsets in St ( A ) witnessing that a given sequence of homomorphisms in H ( A , B ) is notconvergent in some topology.By the Dow–Fremlin theorem and Theorem 8.2 it follows that if A has the inter-polation property, then every pointwise algebraic sequence in H ( A , M κ ) is uniformlyconvergent (Corollary 8.4). Since the interpolation property implies the Grothendieckproperty, we pose the following question being a particular case of Question 9.4. Question 9.6. Assume that A has the (positive) Grothendieck property. Is everypointwise algebraic convergent sequence in H ( A , M κ ) also uniformly convergent?Note that, again by Theorem 8.2, obtaining an algebraic or topological proof foran affirmative answer to Question 9.6 would yield an alternative proof to the theoremof Dow and Fremlin. References [AK92] Edward F. Assmus, Jr. and Jennifer D. Key. Designs and their codes , volume 103 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1992.[BD19] Will Brian and Alan Dow. Small cardinals and efimov spaces. preprint , 2019.[BFH99] Bohuslav Balcar, František Franěk, and Jan Hruška. Exhaustive zero-convergencestructures on Boolean algebras. Acta Univ. Carolinae, Math. et Phys. ,40:27–41, 1999.[BGJ98] Bohuslav Balcar, Wiesław Główczyński, and Thomas Jech. The sequential topology oncomplete Boolean algebras. Fund. Math. , 155(1):59–78, 1998.[BNC] Piotr Borodulin-Nadzieja and Katarzyna Cegiełka. 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Math. , 232(2):501–529, 2019.(Piotr Borodulin-Nadzieja) Instytut Matematyczny, Uniwersytet Wrocławski, pl.Grunwaldzki 2/4, 50-384 Wrocław, Poland Email address : [email protected] (Damian Sobota) Kurt Gödel Research Center for Mathematical Logic, Departamentof Mathematics, University of Vienna, Kolingasse 14-16, 1090 Vienna, Austria Email address ::