Timeline for A result of Sierpiński on non-atomic measures
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| Nov 7, 2015 at 19:09 | comment | added | Salvo Tringali | @Ashutosh. Gerard has probably misread the reference provided by Halmos in relation to Exercise (8) in Sect. 40 of his book (see my 2nd comment above, but change "Sect. 8" to "Sect. 40"). However, I agree with you on the rest: Maharam's characterization (Theorem 1 in her paper) applies only to homogeneous measure algebras, yet Theorem 2 in the same paper yields a decomposition of an arbitrary measure algebra as a (finite or countably infinite) direct sum of homogeneous measure algebras, and this can be used to prove Theorem 1 in the OP. Nonetheless, I'm not sure this answers my question. | |
| Nov 4, 2015 at 22:57 | comment | added | Ashutosh | I haven't looked at Halmos' book but my guess is the following: Maharam's theorem is a characterization of measure algebras - Every atomless measure algebra is essentially a countable join of measure algebras of the product measures on $[0, 1]^{\kappa}$ from which your Theorem 1 immediately follows. | |
| Nov 4, 2015 at 8:06 | comment | added | Salvo Tringali | I've just given a look at Exercise (8) in Sect. 8 of the 1974 Springer ed. of Halmos' Measure Theory. I can't really say at present whether or not it has something to do with the OP, but the ref. provided on p. 291 points to: E. Marczewski, Sur l'isomorphie des mesures séparables, Colloq. Math. 1 (1947), 39-40. Yet, this doesn't match with the db of Colloq. Math., where I could rather find the following: E. Marczewski, Indépendance d'ensembles et prolongement de mesures (Résultats et problèmes), Colloq. Math. 1 (1947), 122-132, which discusses Halmos' exercise in Sect. 1. | |
| Nov 4, 2015 at 7:14 | comment | added | Salvo Tringali | AFAICS, Maharam's paper deals with nontrivial Boolean lattices $\mathbb L = (L, \vee, \land)$ that are closed under countable joins (she calls them "Boolean $\sigma$-algebras") and endowed with a countably additive nonnegative measure $\mu:L\to\bf R$ s.t. $\mu(a)=0$ iff $a=0_{\mathbb L}$, where $0_{\mathbb L}$ is the least element of $\mathbb L$; she refers to $(\mathbb L,\mu)$ as a measure algebra. The paper has 3 theorems and 2 lemmas, which are mostly specialized to homogeneous measure algebras, as defined on the first lines of Section 2. But I don't see any connection with the OP. | |
| Nov 3, 2015 at 22:58 | history | answered | Gerald Edgar | CC BY-SA 3.0 |