3. Let E be a bounded subset of R such that E = U₁1 Ej. Prove that if the E, are of Lebesgue measure zero, then E is Lebesgue measurable and m(E) = 0.
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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.2. Prove the following statements for arbitrary elements of an ordered integral domain . a. If and then . b. If and then . c. If then . d. If in then for every positive integer . e. If and then . f. If and then .For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.
- 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xySuppose that w and r are continuous functions on (−∞, ∞), W (x) is an invertible antiderivative of w(x), and R(x) is an antiderivative of r(x). Circle all of the statements that must be true.
- Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function defined on R, such that for every x it holds that |a(x)| ≤ 1. Using the Global Picard–Lindel¨of Theorem, show that there exists a unique solution y defined on R.Let x, y, z ∈ ℕ. Suppose gcd(x, y) = 1. Prove that if x | yz, then x | z.Let W(t) be the standard Brownian motion, and let X(t) = t W(1/t) for t > 0, X(0) = 0. Show that the covariance (Cov) function of X(t) is the same as the covariance function of W(t): Cov(X(t); X(s)) = Cov(W(t); W(s)) for all s; t > 0. Assuming that the paths of X(t) are continuous with probability 1, argue that X(t) is standard Brownian motion?
- 1 Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over the interval 1 ≤ x ≤ 2 with respect to the weight function r(x) = 1/ x . Obtain the norms of the functions and construct the othornormal set.If E is Lebesgue measurable subset of [a, b], show thatZ baχE = m(E)by using the definition of the Lebesgue integral.Find the Wronskian for the set of functions.{x, ex, sin x, cos x}