Abstract
We study the ultrapowers\(L_1 (\mu )_\mathfrak{U} \) of aL 1(μ) space, by describing the components of the well-known representation\(L_1 (\mu )_\mathfrak{U} = L_1 (\mu _\mathfrak{U} ) \oplus _1 L_1 (\nu _\mathfrak{U} )\), and we give a representation of the projection from\(L_1 (\mu )_\mathfrak{U} \) onto\(L_1 (\mu _\mathfrak{U} )\). Moreover, the subsequence splitting principle forL 1(μ) motivates the following question: if\(\mathfrak{V}\) is an ultrafilter on ℕ and\([f_i ] \in L_1 (\mu )_\mathfrak{V} \), is it possible to find a weakly convergent sequence (g i ) ⊂L 1(μ) following\(\mathfrak{V}\) and a disjoint sequence (h i ) ⊂L 1(μ) such that [f i ]=[g i ]+[h i ]? If\(\mathfrak{V}\) is a selective ultrafilter, we find a positive answer by showing that\(f = [f_i ] \in L_1 (\mu )_\mathfrak{V} \) belongs to\(L_1 (\mu _{_\mathfrak{V} } )\) if and only if its representatives {f i } are weakly convergent following\(\mathfrak{V}\) and\(f \in L_1 (\nu _\mathfrak{V} )\) if and only if it admits a representative consisting of pairwise disjoint functions. As a consequence, we obtain a new proof of the subsequence splitting principle. If\(\mathfrak{V}\) is not a p-point then the above characterizations of\(L_1 (\nu _{_\mathfrak{V} } )\) and\(L_1 (\nu _{_\mathfrak{V} } )\) fail and the answer to the question is negative.
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Supported in part by DGICYT Grant PB 97-0349, Spain.
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González, M., Martínez-Abejón, A. Ultrapowers ofL 1(μ) and the subsequence splitting principle. Isr. J. Math. 122, 189–206 (2001). https://doi.org/10.1007/BF02809899
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DOI: https://doi.org/10.1007/BF02809899