Determining the converge to an element

Reference no: EM13990815

Let (X, S, µ) be a measure space and (S, 1·1) a separable Banach space. A function f from X into S will be called simple if it takes only finitely many values, each on a set in S. Let g be any measurable function from X into S such that ( 1g1 dµ<>∞. Show that there exist simple functions fn with ( 1 fn - g1 dµ → 0 as n → ∞. If f is simple with f (x ) = ),1≤i ≤ n 1A(i )(x )si for si ∈ S and A(i ) ∈ S, let ( f dµ = ),1 ≤ i ≤ n µ( A(i ))si ∈ S. Show that ( f dµ is well-defined and if ( 1 fn - g1 dµ → 0, then ( fn dµ converge to an element of S depending only on g, which will be called ( g dµ (Bochner integral).

Reference no: EM13990815

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