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(8 points) 1. Let /i be continuous functions on the interval...

(8 points) 1. Let /i be continuous functions on the interval [0.1] and suppose that si- / uniformly on the interval. Then, using Lebesgue mensure " and the Lebesgue integral, prove that (8 points) 2. Are the simple functions dense in 1°0(IR) (in the 100 topology)? Why or why not? (...

(1) Suppose that f1 : [0,1] R and f2 : (1, 2] R are measurab...

(1) Suppose that f1 : [0,1] R and f2 : (1, 2] R are measurable functions. Define X € = X € Show that F is a measurable function on [0,2]. (2) Let E C R have the property that m. (EnK) = 0 for all compact subsets K CR. Prove that m. (E) = 0. (3) Given an integrable function f on R, p...

Measure and Probability Lebesgue Integration Let (X,A,) be...

Measure and Probability Lebesgue Integration Let (X,A,) be a measure space and letfi X -R be a measurable nonnegative function, summable against p. Then (as we know) v(A) Sa fu is also a measure on A. Prove that Jgv. of gfu x x for every functiong : X-R summable against v. Hint: start with ...

Measure and Probability Lebesgue Integration (Cont’) ...

Measure and Probability Lebesgue Integration (Cont’) 1. Let µ be a measure defined on the Borel σ - algebra of R. Assume that µ(R) < ∞. Let f(x) be a bounded continuous on R function. Prove that the function g(t)= ∫f (tx)µ(dx), R is continuous o...

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