Borel measurable function from a separable metric

Reference no: EM13966758

1. Let f be a measurable function from X onto S where (X, A) is a measurable space and (S, e) is a metric space with Borel σ-algebra. Let T be a subset of S with discrete relative topology (all subsets of T are open in T ). Show that there is a measurable function g from X onto T . Hint: For f (x ) close enough to t ∈ T , let g(x ) = t ; otherwise, let g(x ) = to for a fixed to ∈ T .

2. Let f be a Borel measurable function from a separable metric space X onto a metric space S with metric e. Show that (S, e) is separable. Hints: As in Problem 8, X has at most c Borel sets. If S is not separable, then show that for some ε> 0 there is an uncountable subset T of S with d(y, z) >ε for all y /= z in T . Use Problem 9 to get a measurable function g from X onto T . All g-1( A), A ⊂ T , are Borel sets in X .

Reference no: EM13966758

Questions Cloud

Convince your audience that short story you are choose : Convince your audience (your instructor and your peers) that the short-short story you are choosing to nominate is the "best" short-short story listed.

Real-valued measurable functions : 1. Let (X, S, µ) be a finite measure space, meaning that µ(X ) ∞. A se- quence { fn } of real-valued measurable functions on X is said to converge in measure to f if for every ε> 0, limn→∞ µ{x : | fn (x ) - f (x )| > ε}= 0. Show that if fn → f in ..

Find the elevation of point q : Write equations in a, b, and c that will ensure that the track is smooth at the transition points - Find the elevation of point Q to find an equation for L2

What happened to the total current supplied : If the switch shown is open, find the real, reactive, and apparent powers in the system. Find the total current supplied to the distribution system by the utility.

Borel measurable function from a separable metric : Let f be a Borel measurable function from a separable metric space X onto a metric space S with metric e. Show that (S, e) is separable.

How do the works of faulkner and beckett relate : How do the works of Faulkner and Beckett relate to existentialism? Are they a continuation, or are they a challenge?

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..

User Account

All Pages