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Measurable Functions with a Given Set of Integrability Exponents

Alfonso Villani
The American Mathematical Monthly
Vol. 118, No. 1 (January 2011), pp. 77-82
DOI: 10.4169/amer.math.monthly.118.01.077
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.118.01.077
Page Count: 6
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Measurable Functions with a Given Set of Integrability
                    Exponents
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Abstract

Abstract Given a measure space (Ω, A, μ), it is well known that for every A-measurable function f : Ω → ℝ the set ℰ(f) = {p ∈ (0, +∞) : f ∈ Lp(μ)} is always an interval, possibly degenerate, but, in general, it cannot be any given interval I ⊆ (0, +∞). Thus we consider the problem of characterizing those measure spaces for which ℰ(f) can be an arbitrary subinterval of (0, +∞). We show that they are precisely the measure spaces such that there is no inclusion between different Lp spaces.

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