Real Analysis in Math

Problem 1 15%

If < fn > is a sequence of measurable functions that converges to a real-valued functions f a.e. on a measurable set E of finite measure, then given η>0, there is a subset A E⊂ with mA<η such that

fn converges to fn uniformly on E A− .

Problem 2. 15%

Problem 3. 30% Convex Function

(1)

(2)

(3)

Problem 4. 20%

Problem 5. 20% For 1≤ <∞p , let f(x) be non-nagative function, find the values of the parameter λ for which