3. Let E be a bounded subset of R such that E=UEj. Provethat if the E, are of Lebesgue measure zero, then E is Lebesguemeasurable and m(E) = 0.4. Let f [a, b] → R be integrable. Define F la hl→ R by

Lebesgue Measure : A set E⊂ℝ is said to be Lebesgue measurable if for every subset of A of ℝ…

Answered: 3. Let E be a bounded subset of R such…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Use the method of undetermined coefficients to determine the form of a particular solution for the given equation. y'"' + 10y" – 11y=xe* + 4x What is the form of the particular solution with undetermined coefficients? Yp(x) = ] (Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.)
11 Prove that if A C R and |A| > 0, then there exists a subset of A that is not Lebesgue measurable. 12 Suppose b < c and A C (b,c). Prove that A is Lebesgue measurable if and only if |A| + |(b,c) \ A| = c – b. 13 Suppose AC R. Prove that A is Lebesgue measurable if and only if |(-n,n) N A| + |(-n,n) \ A| = 2n
2
A set of real numbers D is called dense if and only if every open interval contains at least one point of D. Assume a set E and its complement E = {x € R : x ¢ E} are both dense. Let f : [0, 1] → R be given by f(x) = 1 if x E E and f(x) = 0 if x ¢ E. (Thus f(x) = XE (x) for x E [0, 1].) Prove that f is not integrable on [0, 1].
Let S be a non-empty set of real numbers which is bounded above, and suppose that x = sup(S). (a) Prove that for all € > 0 there exists s ES so that s = (x − €, x]. (b) Suppose now that x S. Prove that for every e > 0 the set {s ES | se (x - e, x]} is infinite.
6.3.4. Theorem The following three properties are equivalent: X is regular. For each xe X and a nbd. U(x) of x there exists another nbd. V(x) of x such that x € V≤veu. (iii) For each x € X and a closed set Ae X not containing or there exists a nbd. V(x) of x such that VA=0. (1) (ii)
Show the following result: Theorem. Assume that f: X → [0, 0] is µ-measurable. Then there exist u-measurable sets {Ak}=1 in X such that 1 f = k=1
Let E and F be measurable sets such that m(EAF) = 0 (where m(EAF) = (E – F) U (F – E)). If ƒ is a measurable function on EUF such that it is integrable on E, then show that f, f = Sr f. E
2. Show that, on the space of continuous functions C[0, 1] the distances defined by the norms = max| |ƒ(x)\, ||f||₁ = √| | \ƒ(1)|dt, ||f||2 = √√√²\ƒ(1)|²³dt, ||f|| = max are mutually non-equivalent.
(c) Show that any countable subset of R is measurable and of Lebesgue measure zero. (d) Use the previous items to justify that if a < b then the interval (a, b) is not countable. (e) Suppose that E is measurable, ECB CR and m* (B) Prove that B is measurable. = m* (E) < ∞.
Suppose A is a Borel measurable subset of [0, 1], m is Lebesgue measure, and ɛ E (0,1). Prove that there exists a continuous function f: [0,1] → R such that 0 <ƒ<1 and m({x: f(x) # XA (x)}) < e.

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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.