Determining the complements of sets

Reference no: EM13966752

Let R be a σ-ring of subsets of a set X . Let S be the σ-algebra generated by R. Recall (§3.3, Problem 8) or prove that S consists of all sets in R and all complements of sets in R.

(a) Let µ be countably additive from R into [0, ∞]. For any set C ⊂ X let µ∗(C ) := sup{µ(B): B ⊂ C, B ∈ R} (inner measure). Show that
µ∗ restricted to S is a measure, which equals µ on R.

(b) Show that the extension of µ to a measure on S is unique if and only if either S = R or µ∗(X \ A) = +∞ for all A ∈ R.

Reference no: EM13966752

Questions Cloud

Measurable real-valued function : Let (X, S) be a measurable space and En measurable sets, not necessarily disjoint, whose union is X. Suppose that for each n, fn is a measurable real-valued function on En. Suppose that for any x ∈ Em ∩ En for any m and n, fm (x ) = fn (x ). Le..

How would you provide constructive feedback : How would you provide constructive feedback to this person without causing any ill will between the two of you? Evaluate your options according to what you feel to be your chances for success.

Problem regarding the equivalence relation : Define a relation x ≡ u iff for all n, x =n u. Show that this is an equivalence relation which has at most c different equivalence classes, and for any x, y, and u, if x ≡ u, then (x, y) ∈ C if and only if (u, y) ∈ C . For C = D and y = x , find a..

Second-countable topological space : Let (S, T ) be a second-countable topological space and (Y, d) any metric space. Show that the Borel σ-algebra in the product S × Y is the product σ-algebra of the Borel σ-algebras in S and in X.

Determining the complements of sets : Let R be a σ-ring of subsets of a set X . Let S be the σ-algebra generated by R. Recall (§3.3, Problem 8) or prove that S consists of all sets in R and all complements of sets in R.

Provide directions for your examples : What step would you add? Provide 2 examples for your classmates to attempt. Provide directions for your examples.

Find the acceleration at seconds : Suppose that an automobile's velocity starting from rest is v(t) = 270t/3t + 11 where v is measured in feet per second. Find the acceleration at 7 seconds. Round your answer to one decimal place.

Problem regarding the reversed martingale : Let Ck be the smallest σ-algebra for which Uj is measurable for all j ≤ k. Show that {Uk, Ck }k≤-1 is a reversed martingale and find its limit as k → -∞. Let C-∞ := nk≤-1 Ck . Is Y equal a.s. to a variable measurable for C-∞?

What is the total line current il when the switch is open : The figure shown below shows a one-line diagram of a simple power system containing a single 480 V generator and three loads. Assume that the transmission lines in this power system are lossless, and answer the following questions. Assume that..

User Account

All Pages