This paper is devoted to hyperstonean spaces that are precisely the Stone spaces of measure algebras, or the Stone spaces of the Boolean algebras of Lp-projections of Banach spaces for 1 ≤ p < ∞, p ̸= 2. Several new results that have been achieved recently are discussed. | Turk J Math (2018) 42: 2288 – 2295 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article On some properties of hyperstonean spaces Banu GÜNTÜRK∗,, Bahaettin CENGİZ, Faculty of Engineering, Başkent University, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: This paper is devoted to hyperstonean spaces that are precisely the Stone spaces of measure algebras, or the Stone spaces of the Boolean algebras of Lp -projections of Banach spaces for 1 ≤ p 0} and let F0 = (F ∩ Ωj ). Then the set F \ F0 has finite measure and j∈J as in the proof of the preceding theorem its intersection with each Ωi has measure zero, which means that is F is contained µ . in the union of the countable subfamily {Ωj : j ∈ I} of G. ∪ Now suppose that Z = Fk where Fk s are mutually disjoint and have finite measure. k≥1 By the above discussion, for each k, Fk is contained almost everywhere in the union of countable ∪ Gk of G. subfamily Gk of G , which implies that Z is contained . in the union of the countable subfamily Thus, G = ∪ k Gk is countable. k≥1 The converse is trivial, for if G = {Ωk : k = 1, 2, 3, . . .} is a clopen µ decomposition of µ then ∪ Ωk } is a countable partition of Z of sets with finite measure. 2 G ∪ {Z \ The following theorem establishes a relation between the σ -finiteness of µ and the existence of a finite perfect measure. Theorem 4 µ is σ -finite if and only if there exists a perfect regular Borel measure on .
