Measure and the fn are uniformly integrable

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For a finite measure space (X, S, µ), a set F of integrable functions is said to be uniformly integrable iff sup{( | f | dµ: f ∈ F } <>∞ and for every ε > 0, there is a δ > 0 such that if µ( A) <>δ, then (A | f | dµ <>ε for every f ∈ F . Prove that a sequence { fn } of integrable functions satisfies ( | fn - f | dµ → 0 if and only if both fn → f in measure and the fn are uniformly integrable.

Reference no: EM13990760

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