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solns9b - Mathematics 105 Spring 2004 M Christ Problem Set...

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Mathematics 105 — Spring 2004 — M. Christ Problem Set 9 — Solutions to Selecta, part 2 3.3.21(i) Let K be a family of measurable functions on a measure space ( E, B , μ ). Show that K is uniformly μ -integrable if it is uniformly μ -absolutely continuous and satisfies sup f ∈K f L 1 ( μ ) < . Conversely, show that if K is uniformly μ -integrable then it is uniformly μ -absolutely continuous. Show that if in addition μ ( E ) < , then sup f ∈K f L 1 ( μ ) < . Solution. Suppose that K is uniformly μ -absolutely continouus and satisfies sup f ∈K f L 1 ( μ ) = C < . For any positive number R < and any f ∈ K , μ ( { x : | f ( x ) | ≥ R } ) R - 1 f L 1 CR - 1 by Markov’s inequality. Let ε > 0. By hypothesis there exists δ > 0 such that A | f | dμ < ε for all A ∈ B satisfying μ ( A ) < δ . If R is chosen to be sufficiently large that C/R < δ then the set A = { x : | f ( x ) | ≥ R } satisfies μ ( A ) < δ and hence { x : | f ( x ) |≥ R } | f | dμ < ε . Conversely suppose that K is uniformly μ -integrable. For any measurable set A with finite measure, A | f | A ∩{| f | >R } |
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