×

Completeness of \(L_1\) spaces over finitely additive probabilities. (English) Zbl 0940.28005

Various necessary and sufficient conditions for completeness of the \(L_1\) space of a finitely additive measure on a field of sets are known. The authors use these conditions to study completeness of the \(L_1\) space of a finitely additive probability measure on a \(\sigma\)-field in relation to the countably additive, strongly continuous and discrete components of the measure. In this setting, \(\gamma \) is a bounded finitely additive measure on a \(\sigma\)-field \({\mathcal F}\) of subsets of a set \(\Omega\). By the Hewitt-Yosida and Sobczyk-Hammer decomposition theorems, \(\gamma = \gamma_0 + \gamma_1 = \gamma_2 + \gamma _3 \) where \(\gamma_0 \) is countably additive, \(\gamma_1 \) is purely finitely additive, \(\gamma_2\) is discrete and \(\gamma_3\) is strongly continuous. Then \(\gamma_2 = \sum_{j \in J}a_j \delta_j\) where \(J\) is a subset of the positive integers, \(\{\delta_j \}\) is a collection of distinct 0-1 measures, and \(\sum a_j\) is a convergent series of nonnegative terms. It is proved that (1) \(L_1(\gamma)\) is complete if and only if \(L_1(\gamma_1)\) is complete and \(\gamma_0\) and \(\gamma_1 \) are mutually singular; (2) \(L_1(\gamma)\) is complete if and only if \(L_1(\gamma_2)\) and \(L_1(\gamma_3)\) are complete and the two measures are mutually singular; and (3) for any bounded discrete finitely additive measure in the form of \(\gamma_2\) above, \(L_1(\gamma_2)\) is complete if and only if \(\{\delta_j \}\) is uniformly singular, that is, there is a measurable partition \(\{A_j \}\) of \(\Omega\) such that \(\delta_j(A_j) =1\) for each \(j \in J\). This leads to the theorem: \(L_1(\gamma)\) is complete if and only if there is a measurable partition \(\{A,B,C\}\) of \(\Omega\) such that \(\gamma\) is countably additive on \(A\); \(\gamma\) is discrete on \(B\) and, when this discrete component is expressed in the form of \(\gamma_2\) above, \(\{\delta_j \}\) is uniformly singular; \(\gamma\) is strongly continuous on \(C\); and \(L_1(C,\gamma_C)\) is complete. Following L. E. Dubins and L. J. Savage [“Inequalities for stochastic processes. How to gamble if you must” (1976; Zbl 0359.60002)], the authors also consider finite strategic products of probability spaces \((\Omega_i , {\mathcal P} (\Omega_i), \gamma_i)\). Completeness of \(L_1(\gamma_i)\) for each \(i\) guarantees completeness of the product \(L_1\) space, but the converse fails: only the first factor space \(L_1(\gamma_1)\) is certain to be complete if the product \(L_1\) space is complete. Partial results and open questions are stated for infinite strategic products. Several useful examples are also provided.

MSC:

28A25 Integration with respect to measures and other set functions
60A10 Probabilistic measure theory
28A35 Measures and integrals in product spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Citations:

Zbl 0359.60002
PDFBibTeX XMLCite
Full Text: DOI EuDML