Real-valued measurable functions

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1. Let (X, S, µ) be a finite measure space, meaning that µ(X ) ∞. A se- quence { fn } of real-valued measurable functions on X is said to converge in measure to f if for every ε> 0, limn→∞ µ{x : | fn (x ) - f (x )| > ε}= 0. Show that if fn → f in measure and for some integrable function g, | fn |≤ g for all n, then ( | fn - f | dµ → 0.

2. (a) If fn → f in measure, fn ≥ 0 and ( fn dµ → ( f dµ ∞, show that ( | fn - f | dµ → 0. Hint: See Problem 1.

(b) If fn → f in measure and ( | fn | dµ → ( | f | dµ <>∞, show that ( | fn - f | dµ → 0.

Reference no: EM13966761

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