Let Ω be any set and let F be the collection of all subsets of Ω...

Let Ω be any set and let F be the collection of all subsets of Ω that are either countable or have a countable complement. (Recall that a set is countable if it is either finite or can be placed in one-to-one correspondence with the natural numbers N = {1, 2, . . .}.)

(a) Show that F is a σ-algebra.

(b) Show that the set function given by

μ(E)= 0 if E is countable ;

μ(E) = ∞ otherwise

is a measure, where E ∈ F.

Expert Solution

Colby Messinger answered 2 years ago

Let (Ω, F , P) be a probability space. Suppose that Ω is the collection of...

Let (Ω, F , P) be a probability space. Suppose that Ω is the collection of all possible outcomes of a single iteration of a certain experiment. Also suppose that, for each C ∈ F, the probability that the outcome of this experiment is contained in C is P(C). Consider events A, B ∈ F with P(A) + P(B) > 0. Suppose that the experiment is iterated indefinitely, with each iteration identical and independent of all the other iterations, until...

Let S be a set of n numbers. Let X be the set of all subsets...

Let S be a set of n numbers. Let X be the set of all subsets of S of size k, and let Y be the set of all ordered k-tuples (s1, s2, , sk) such that s1 < s2 < < sk. That is, X = {{s1, s2, , sk} | si S and all si's are distinct}, and Y = {(s1, s2, , sk) | si S and s1 < s2 < < sk}. (a) Define a one-to-one correspondence f : X → Y. Explain...

Definition 1 (Topological space). Let X be a set. A collection O of subsets of X...

Definition 1 (Topological space). Let X be a set. A collection O of subsets of X is called a topology on the set X if the following properties are satisfied: (1) emptyset ∈ O and X ∈ O. (2) For all A,B ∈ O, we have A∩B ∈ O (stability under intersection). (3) For all index sets I, and for all collections {Ui}i∈I of elements of O (i.e., Ui ∈ O for all i ∈ I), we have U i∈I...

1. Suppose ?:? → ? and {??}?∈? is an indexed collection of subsets of set ?....

1. Suppose ?:? → ? and {??}?∈? is an indexed collection of subsets of set ?. Prove ?(⋂ ?? ?∈? ) ⊂ ⋂ ?(??) ?∈? with equality if ? is one-to-one. 2. Compute: a. ⋂ ∞ ?=1 [?,∞) b. ⋃ ∞ ?=1 [0,2 − 1 /?] c. lim sup ?→∞ (−1 + (−1)^? /?,1 +(−1)^? /?) d. lim inf ?→∞(−1 +(−1)^?/ ?,1 +(−1)^? /?)

Let X be the set of all subsets of R whose complement is a finite set...

Let X be the set of all subsets of R whose complement is a finite set in R: X = {O ⊂ R | R − O is finite} ∪ {∅} a) Show that T is a topological structure no R. b) Prove that (R, X) is connected. c) Prove that (R, X) is compact.

Let f : [0, 1] → R and suppose that, for all finite subsets of [0,...

Let f : [0, 1] → R and suppose that, for all finite subsets of [0, 1], 0 ≤ x1 < x2 < · · · < xn ≤ 1, we have |f(x1) + f(x2) + · · · + f(xn)| ≤ 1. Let S := {x ∈ [0, 1] : f(x) ̸= 0}. Show that S is countable

2. Let A = {1,2,3,4}. Let F be the set of all functions from A to...

2. Let A = {1,2,3,4}. Let F be the set of all functions from A to A. Recall that IA ∈ F is the identity function on A given by IA(x) = x for all x ∈ A. Consider the function E : F → A given by E(f) = f(1) for all f ∈ F. (a) Is the function E one-to-one? Prove your answer. (b) Is the function E onto? Prove your answer. (c) How many functions f ∈...

Consider the following subsets of the set of all students: A = set of all science...

Consider the following subsets of the set of all students: A = set of all science majors B = set of all art majors C = set of all math majors D = set of all female students Using set operations, describe each of the following sets in terms of A, B, C, and D: a) set of all female physics majors b) set of all students majoring in both science and art

Let {Kn : n ∈ N} be a collection of nonempty compact subsets of R N...

Let {Kn : n ∈ N} be a collection of nonempty compact subsets of R N such that for all n, Kn+1 ⊂ Kn. Show that K = T∞ n=1 Kn is compact. Can K ever be the empty set?

c) Let R be any ring and let ??(?) be the set of all n by...

c) Let R be any ring and let ??(?) be the set of all n by n matrices. Show that ??(?) is a ring with identity under standard rules for adding and multiplying matrices. Under what conditions is ??(?) commutative?

Question