Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with ξ ∈ (0, 1) (note that the case ξ = 1/3 corresponds to the regular Cantor set we learned in our lectures). In stage 1 of the construction, remove the centrally situated open interval in [0, 1] of length ξ. In stage 2 remove the centrally situated open intervals each of relative length ξ (i.e. if the interval has length a you remove an interval of length ξ × a), one in each of the remaining intervals after stage 1, and so on. Let C_ξ denote the set which remains after applying the above procedure indefinitely

(a) Prove that C_ξ is compact.

(b) Prove that C_ξ is totally disconnected and perfect.

(c) Atually, prove that the complement of C_ξ in [0, 1] is the union of open intervals of total length equal to 1.

(d) Show directly that m_∗(C_ξ) = 0.

Expert Solution

Colby Messinger answered 1 year ago

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with ξ ∈ (0, 1) (note that the case ξ = 1/3 corresponds to the regular Cantor set we learned in our lectures). In stage 1 of the construction, remove the centrally situated open interval in [0, 1] of length ξ. In stage 2 remove the centrally situated open intervals each of relative length ξ (i.e. if the interval has length a you remove an...

6.1.5. Problem. Let J be the open unit interval (0, 1). For each a let Ua...

6.1.5. Problem. Let J be the open unit interval (0, 1). For each a let Ua = ?a, a + 1 ?, and let U = {Ua : 0 ≤ a ≤ 34 }. Then certainly U covers J . (a) Find a finite subfamily of U which covers J. (b) Explain why a solution to (a) does not suffice to show that J is compact. (c) Show that J is not compact.

A. Prove that R and the real interval (0, 1) have the same cardinality.

2. Let X be a uniform random variable over the interval (0, 1). Let Y =...

2. Let X be a uniform random variable over the interval (0, 1). Let Y = X(1-X). a. Derive the pdf for Y . b. Check the pdf you found in (a) is a pdf. c. Use the pdf you found in (a) to find the mean of Y . d. Compute the mean of Y by using the distribution for X. e. Use the pdf of Y to evaluate P(|x-1/2|<1/8). You cannot use the pdf for X. f. Use...

Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]. Let...

Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]. Let ?? be the ?-th smallest among {?1, ?2, ?3}. Find the variance of ?2, and the covariance between the median ?2 and the sample mean ? = 1 3 (?1 + ?2 + ?3).

Let U be a continuous uniform variable over the interval [0, 1]. What is the probability...

Let U be a continuous uniform variable over the interval [0, 1]. What is the probability that U falls within kσ of its mean for k = 1, 2, 3?

Real numbers p and q are randomly chosen from the interval 0 to 1, inclusive. If...

Real numbers p and q are randomly chosen from the interval 0 to 1, inclusive. If r is given by r = 2(p + q), and p, q, r are rounded to the nearest integers to give P, Q and R, respectively, determine the probability that R = 2(P + Q). (As an example, if p=0.5 and q=0.381, then r=1.762, and so P =1, Q=0, and R=2)

Days 1 to 120: Number of New Cases 0 1 0 0 2 0 3 0...

Days 1 to 120: Number of New Cases 0 1 0 0 2 0 3 0 0 0 0 14 2 1 0 27 80 51 18 26 216 81 37 117 167 108 151 178 111 414 337 195 706 214 300 68 118 160 190 209 154 193 170 154 166 68 119 152 146 101 98 159 175 195 195 157 130 129 161 116 133 217 209 75 298 240 205 196 233 76 339 170...

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be the usual “less than or equal to” total order on the set A=(−2,0)∪(0,2) Consider the subset B={−1/n | n∈N, n≥1}⊆A. Determine an upper bound for B in A. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound. Note: least upper bound is an upper...

2. Let x be a real number, and consider the deleted neighborhood N∗(x;ε). (a) Show that...

2. Let x be a real number, and consider the deleted neighborhood N∗(x;ε). (a) Show that every element of N∗(x;ε) is an interior point. (b) Determine the boundary of N∗(x;ε) and prove your answer is correct.

Question

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

Solutions

Expert Solution

Related Solutions

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

6.1.5. Problem. Let J be the open unit interval (0, 1). For each a let Ua...

A. Prove that R and the real interval (0, 1) have the same cardinality.

2. Let X be a uniform random variable over the interval (0, 1). Let Y =...

Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]. Let...

Let U be a continuous uniform variable over the interval [0, 1]. What is the probability...

Real numbers p and q are randomly chosen from the interval 0 to 1, inclusive. If...

Days 1 to 120: Number of New Cases 0 1 0 0 2 0 3 0...

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...

2. Let x be a real number, and consider the deleted neighborhood N∗(x;ε). (a) Show that...