Let u* be an outer measure on P(X) and M be the collection of all measurable sets in P(X). Let (An) be a sequence of pairwise disjoint sets in M. Then show that for any T € P(X), An I H° (TN A,.). n=1
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- Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.Label each of the following statements as either true or false. Let f:AB where A and B are nonempty. Then f1(f(T))=T for every subset T of B.25. Let, where and are non empty, and let and be subsets of . Prove that. Prove that. Prove that. Prove that if.
- Let (X, B) be a measurable space and {μn} a sequence ofmeasures with the property that for every E ∈ B,μn(E) ≤ μn+1(E), n = 1, 2,... .Let μ(E) = limn→∞ μn(E). Show that (X, B, μ) is a measurespace.Let S be a nonempty set of real numbers that is bounded from above (below) and let x = sup S (inf S). Prove that either x belongs to S or x is an accumulation point of S.Let f: [a, b] -> R be continuous except on a finite number of points in (a, b]. Is f Lebesgue measurable? Answer it Yes or No. True or False: If f: E - › R is continuous and E has a Lebesgue measure 0, then its image f(E) has a Lebesgue measure 0.
- Let f : a) Prove that if f is measurable on E and f = g a.e. on E, then g is measurable on E. b) For a measurable subset D of E, f is measurable on E if only if the restriction of f to D and ED are measurable. - R be a function on a measurable set E.Find an example of a bounded convex set S in R2 such that its profile P is nonempty but conv P ≠ S.Let f ∈ L[a, b]. Show that if g is a bounded measurablefunction, then fg ∈ L[a, b].
- Let E ⊂ ℝ and y ∈ ℝ. Show that E + y := {x + y : x ∈ E} is measurable then E ismeasurable. Give a complete and detailed proof.Let E ⊂ ℝ and y ∈ ℝ. Show that E is measurable iff E + y := {x + y : x ∈ E} ismeasurable. Give a complete and detailed proof.If f is continuous on the closed interval [a, b], which of the following must be true: (a) f has at least one critical point on (a, b)(b) there exists a point c ∈ (a,b) such that f(c) = 0(c) f′(c) exists for all c ∈ (a,b) (d) f has an absolute maximum and absolute minimum on [a, b] (e) both (a) and (b) are true.